Download PDF

Journal of Homotopy and Related Structures

, Volume 8, Issue 2, pp 151–200 | Cite as

Shadows and traces in bicategories

Article
First Online:
Received:
Accepted:

Abstract

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative” traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow”. In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate “cylindrical” type of string diagram, which we justify formally in an appendix.

Keywords

Bicategory Trace Fixed-point theory 

Mathematics Subject Classification (2000)

18D05 55M20 

1 Introduction

The purpose of this paper is to study and exposit a categorical notion of trace for endo-2-cells in a bicategory. Since there are also other sorts of categorical “traces,” we begin by briefly describing where ours fits into the general picture. Probably the most basic sort of trace is the trace of a square matrix over a field. This generalizes to square matrices over a commutative ring, and even to endomorphisms of finitely generated projective modules over a commutative ring. It is well-known that there is an appropriate categorical definition of such traces, which applies to any endomorphism of a dualizable object in a symmetric monoidal category; see, for example [8, 18].

This general definition includes many important notions in topology, geometry, and algebra. For instance, traces in the stable homotopy category can be identified with fixed point indices, which include in particular Euler characteristics (as the traces of identity maps)—while traces in the derived category of a ring are called Lefschetz numbers. Moreover, the category-theoretic definition of trace trivially implies that it is preserved by symmetric monoidal functors, such as homology. We thereby immediately obtain the classical Lefschetz fixed point theorem: if an endomorphism of a dualizable space (such as a finite-dimensional manifold) has no fixed points, then its fixed point index is zero, and hence so is the Lefschetz number of the map it induces on homology. The search for similar category-theoretic expressions of other fixed-point invariants was the primary motivation for the generalization of trace we will present in this paper; see [28, 29, 30].

One way to generalize traces is to remove the requirement that the objects be dualizable. In this case, traces become extra structure imposed on a symmetric monoidal category; see [17]. Such traces can even exist in cartesian monoidal categories, though there are no nontrivial dual pairs in such a category. The connection with fixed points is then even more striking: a trace on a cartesian monoidal category is equivalent to an operator which chooses a fixed point for every map (see [10]). These sort of traces are especially important in computer science, where the fixed-point operator is identified with a recursion combinator. However, they are not particularly useful in topological fixed point theory, since a given map may have zero, one, two, or more fixed points of equal importance, and so there can be no uniform fixed-point–assigning operator.

Thus, we will generalize in a different direction: we still consider only dualizable objects, but relax the requirement of symmetry. It is easy to define traces for dualizable objects in a monoidal category which is merely braided, or more precisely balanced (see, e.g. [17]). However, the applications in [28, 29, 30] require notions of trace in even less commutative situations. For instance, Stallings [37] proved that for modules over a noncommutative ring, there is a unique notion of trace which is additive and cyclic. This “Hattori-Stallings trace” (see also [11]) lives in a suitable quotient of the ring, rather than the ring itself. We will present a generalization of the symmetric monoidal trace which includes the Hattori-Stallings trace. It was invented by the first author for applications in fixed point theory; see [29].

Modules over noncommutative rings do not form a monoidal category at all, but rather a bicategory (or, if necessary, a double category; see e.g. [36]). It makes no sense to ask whether a bicategory is symmetric, braided, or balanced, but it turns out that there is a type of structure we can impose on a bicategory which enables us to define traces therein. We refer to this structure as a shadow, although it could also quite reasonably be called a “2-trace,” since it is itself a categorified kind of trace. (This is an example of the microcosm principle of [2].) Many bicategories are naturally equipped with shadows, such as the following:
  1. (i)

    Rings and bimodules (the Hattori-Stallings context), as well as generalizations such as DGAs and chain complexes.

     
  2. (ii)

    Parametrized spaces and parametrized spectra, as studied in [22].

     
  3. (iii)

    Spaces with group actions, as studied in [33].

     
  4. (iv)

    \(n\)-dimensional manifolds and cobordisms.

     
  5. (v)

    Categories and profunctors (also called bimodules or distributors), with enriched and internal variations.

     
The goal of this paper is to define and explain the notions of shadow and trace in bicategories and prove some of their basic properties. The main property we are interested in is functoriality, since this is what makes Lefschetz-style theorems fall out easily. Thus, the main part of the paper can be regarded as a build-up to our functoriality results.

Our intent is to make this paper a “bridge”, accessible to two audiences. On the one hand, there are topologists familiar with the topological applications and interested in the formal foundations who may appreciate an introduction to the basic category-theoretic ideas. On the other hand, there may be category-theorists interested in applications of categorified traces, who will understand the categorical ideas and notation already, but may have little background in topology. This makes for a difficult balancing act.

Since the central definitions and results in this paper are category-theoretic, we have chosen to spend more time on the necessary categorical background. The topology appears only in examples, and for these we give some intuitive description, along with references for further reading. We have also tried to include enough non-topological examples that even a reader without any topological background can grasp the categorical ideas—although we stress that the most important applications are topological (see [28, 29, 30, 32]).

We start with a brief review of the classical theory of symmetric monoidal traces, so that the analogies with bicategorical trace will be clear. Section 2 contains the basic definitions, examples, and properties, including the all-important functoriality. Less often cited, but also important, is what we call “2-functoriality”: symmetric monoidal traces commute not only with functors but with natural transformations. Everything in this section can be found in classical references such as [8, 17]. We also summarize the classical “string diagram” notation for symmetric monoidal categories, which provides a convenient notation and calculus for manipulating composites of many morphisms; see [15, 17, 25, 26, 34, 39]. Section 3 is also a review of classical material, this time the definition of bicategories and their string diagrams.

The next Sects. 4 and 5 contain the basic definitions of shadows and traces, respectively, with Sect. 6 devoted to a number of examples. These definitions originally appeared in [29], but here we study them carefully from a categorical perspective. In particular, in Sect. 7 we prove a number of formal properties of the bicategorical trace, analogous to the familiar formal properties of symmetric monoidal trace. These properties are most conveniently expressed and proven using an appropriate string diagram calculus, which in the case of shadows involves diagrams drawn on a cylinder; we introduce these string diagrams in Sect. 4.

Finally, in Sects. 8 and 9 we prove the crucial results about functoriality and 2-functoriality for the bicategorical trace, starting with the necessary definitions. Just as a bicategory must be equipped with the extra structure of a shadow in order to define traces, a functor of bicategories must be given the structure of a “shadow functor” in order for it to respect traces. The appropriate notion of “shadow transformation” is somewhat more subtle, involving a bicategorical transformation whose components are dual pairs, rather than single 1-cells. The definitions of shadow functor and shadow transformation are included in Sects. 8 and 9 respectively.

Finally, in Appendix  we give a formal basis to our cylindrical string diagrams by proving that any such labeled string diagram determines a unique deformation-invariant composite. This is a technical, but fairly straightforward extension of the classical proof for monoidal categories in [15].

2 Traces in symmetric monoidal categories

We begin by reviewing traces in symmetric monoidal categories, using string diagram calculus, a summary of which can be found in Fig. 1. These diagrams were first used by Penrose [25, 26], given a rigorous foundation by Joyal and Street [15, 17], and since then have been adapted to many different contexts; a comprehensive overview can be found in [34]. They may be called “Poincaré dual” to the usual sorts of diagrams: instead of drawing objects as vertices and morphisms as arrows, we draw objects as strings and morphisms as vertices, often with boxes around them. We may think of a morphism as a “machine” with its domain drawn as “input” strings coming into it and its codomain as “output” strings going out of it.
Fig. 1

String diagrams for monoidal categories

All of our string diagrams are read from top to bottom. After a while we will omit the arrowheads on the strings, but in the beginning they can help to clarify the intent. Note that although we draw the symmetry \(M \otimes N \cong N \otimes M\) with one string crossing “over” the other, there is no meaning assigned to which one is in front, since our monoidal category is symmetric and not merely braided.

Joyal and Street [15] gave a formal definition of the “value” of a “string diagram” whose arrows and vertices are labeled by objects and morphisms in a monoidal category, and showed that this value is invariant under deformations of diagrams. Thus, manipulation of string diagrams is actually a fully rigorous way to prove theorems about symmetric monoidal categories.

Definition 2.1

Let C be a symmetric monoidal category with product \(\otimes \) and unit object \(I\). An object \(M\) of C is dualizable if there exists an object \({{M}^{\bigstar }}\), called its dual, and maps
$$\begin{aligned} I \overset{\eta }{\longrightarrow } M\otimes {{M}^{\bigstar }} \qquad \qquad {{M}^{\bigstar }}\otimes M \overset{\varepsilon }{\longrightarrow } I \end{aligned}$$

satisfying the triangle identities \((\text{ id}_M \otimes \varepsilon )(\eta \otimes \text{ id}_M) \,{=}\, \text{ id}_M\) and \((\varepsilon \otimes \text{ id}_{M^{\star }}) (\text{ id}_{M^{\star }} \otimes \eta ) = \text{ id}_{M^{\star }}\). We call \(\varepsilon \) the evaluation and \(\eta \) the coevaluation.

Note that any two duals of an object \(M\) are canonically isomorphic, and that if \({{M}^{\bigstar }}\) is a dual of \(M\), then \(M\) is also a dual of \({{M}^{\bigstar }}\). If \(M\) and \(N\) are dualizable and \(f:Q\otimes M \rightarrow N\otimes P\) is a morphism in \(\mathbf C \), the mate of \(f\) is the composite
$$\begin{aligned}&{{N}^{\bigstar }}\otimes Q\xrightarrow {{{\text{ id}}\; \otimes \;\eta }}\ {{N}^{\bigstar }}\otimes Q\otimes M\otimes {{M}^{\bigstar }} \quad \\&\quad \xrightarrow {{{\text{ id}}\; \otimes f\otimes \text{ id} }} {{N}^{\bigstar }}\otimes N\otimes P\otimes {{M}^{\bigstar }} \xrightarrow {{\varepsilon \otimes \;\text{ id} }} \ P\otimes {{M}^{\bigstar }} \end{aligned}$$
Dual objects are represented graphically by turning around the direction of arrows, while the triangle identities translate into “bent strings can be straightened;” see Figs. 2 and 3. String diagrams for monoidal categories with duals are formalized in [13, 14] allowing arbitrary behavior of strings (not restricting them to travel only vertically). However, for most purposes it suffices to consider only vertical strings, regarding each “turning around” of a string as a vertex implicitly labeled by \(\eta \) or \(\varepsilon \) and the triangle identities simply as axioms (rather than deformations).
Fig. 2

Coevaluation and evaluation

Fig. 3

The triangle identities

We now recall the definition of the trace of an endomorphism of a dualizable object in a symmetric monoidal category. In fact, we can take the trace of more than just endomorphisms; we only require the dualizable object to appear as a factor of the source and target.

Definition 2.2

Let \(\mathbf C \) be a symmetric monoidal category, \(M\) a dualizable object of \(\mathbf C \), and \(f:Q\otimes M\rightarrow M\otimes P\) a morphism in \(\mathbf C \). The trace \(\mathrm{tr }(f)\) of \(f\) is the following composite:
$$\begin{aligned} Q \overset{\eta }{\longrightarrow } Q\otimes M\otimes {{M}^{\bigstar }} \overset{f}{\longrightarrow } M \otimes P \otimes {{M}^{\bigstar }} \mathop {\longrightarrow }\limits _{\sim }^\mathfrak{s }{{M}^{\bigstar }}\otimes M\otimes P \overset{\varepsilon }{\longrightarrow } P\qquad \end{aligned}$$
(2.1)
 

References for this notion of trace include [8, 9, 16, 17, 18]. The definition does not depend on the choice of dual \({{M}^{\bigstar }}\) or the evaluation and coevaluation maps. When \(Q=P=I\), it reduces to the more familiar definition of the trace of an endomorphism. There are also two other degenerate cases of this definition that classically go by other names: the Euler characteristic (or dimension) of \(M\) is the trace of its identity map, and the transfer of \(M\) is the trace of a “diagonal” morphism \(\Delta :M\rightarrow M\otimes M\) when such exists (so in this case \(P=M\) and \(Q=I\)).

In string diagram notation, the trace of a morphism looks like “feeding its output into its input;” see Fig. 4. The second picture looks cleaner and is more commonly drawn, but we note that it makes essential use of the symmetry, in regarding \(f\) as a morphism \(Q\otimes M \rightarrow P\otimes M\) and in switching the order in the source of the evaluation map. Thus, since we intend to generalize away from symmetry, it is important to keep the first picture in mind as well.
Fig. 4

The symmetric monoidal trace

 

Example 2.3

Let \(\mathbf C =\mathbf Vect _{k}\) be the category of vector spaces over a field \(k\). A vector space is dualizable if and only if it is finite-dimensional, and its dual is the usual dual vector space. We have \(I=k\) and \(\mathbf C (I,I)\cong k\) by multiplication. Using this identification, Definition 2.2 recovers the usual trace of a matrix. The Euler characteristic of a vector space (i.e. the trace of its identity map) is its dimension.

This example generalizes to modules, chain complexes of modules, and the derived category of modules over a commutative ring. In these cases the dualizable objects are the finitely generated projectives, finitely generated chain complexes of projectives, and chain complexes quasi-isomorphic to a finitely generated chain complex of projectives respectively.

If \(V= k[S]\) is the free module on a finite set \(S\), then the diagonal \(S\rightarrow S\times S\) induces a diagonal \(V\rightarrow V\otimes V\). Thus, any endomorphism of \(V\) has a “transfer” \(k\rightarrow V\), which just picks out an element of \(V\) (the image of 1). If \(f:V\rightarrow V\) is induced by an endomorphism \(\hat{f}:S\rightarrow S\), then its transfer is the sum \(\sum _{\hat{f}(s)=s} s\) of all the fixed points of \(\hat{f}\).

 

Example 2.4

Another important class of examples is topological: whenever \(M\) is a closed smooth manifold (or a compact ENR), its suspension spectrum \(\Sigma ^\infty (M_+)\) is dualizable in the stable homotopy category. This is called Spanier-Whitehead duality.

For the reader unfamiliar with stable homotopy theory, what this means in concrete terms is that there is a pointed space \(N\), together with evaluation and coevaluation maps
$$\begin{aligned} S^n \overset{\eta }{\longrightarrow } M_+ \wedge N N \wedge M_+ \overset{\varepsilon }{\longrightarrow } S^n \end{aligned}$$
for some large-dimensional sphere \(S^n\), such that the triangle identities commute up to “stable homotopy”. Working “stably” means roughly that we can do and undo smash products with spheres freely, and the stable homotopy category is a way of making that precise. When formulated explicitly as above, this sort of duality is called \(n\)-duality.

Now, given an endomorphism \(f\) of \(M\), we can either map it into the stable homotopy category and take its trace as an endomorphism of \(\Sigma ^\infty (M_+)\), or define a trace directly using \(n\)-duality. In the latter case, we obtain an endomorphism of \(S^n\), which is classified by an integer (its degree). This is the fixed point index of \(f\), which counts the number of fixed points of \(f\) with multiplicity; see [6, 7, 8, 20, 22]. In particular, the Euler characteristic of \(\Sigma ^\infty (M_+)\) (that is, the trace of its identity map) can be identified with the usual Euler characteristic of \(M\).

Here we also have a diagonal \(\Sigma ^\infty (M_+) \rightarrow \Sigma ^\infty (M_+) \wedge \Sigma ^\infty (M_+)\) induced by the diagonal \(\Delta :M\rightarrow M\times M\) of \(M\), and hence there is a transfer \(S\rightarrow \Sigma ^\infty (M_+)\). This transfer can again be regarded as the “formal sum” of all the fixed points of the original endomorphism.

 

Remark 2.5

Note that the functors \(k[-]:\mathbf{Set } \rightarrow \mathbf{Vect _{k}}\) and \(\Sigma ^{\infty }(-)_{+}:\mathbf{Top } \rightarrow \mathbf{Sp }\) play a similar role in Examples 2.3 and 2.4. In both cases we start with an object in a cartesian monoidal category, where there are no nontrivial dualizable objects or traces, and apply a functor landing in a noncartesian monoidal category, after which our object becomes dualizable and we can calculate traces. Moreover, in both cases the noncartesian monoidal category is “additive” and the trace gives us the “sum” of all the fixed points of a map in our original cartesian monoidal category.

The symmetric monoidal trace has the following fundamental property.

Proposition 2.6

(Cyclicity) If \(M\) and \(N\) are dualizable and \(f:Q\otimes M\rightarrow N\otimes P\) and \(g:K\otimes N\rightarrow M\otimes L\) are morphisms, then
$$\begin{aligned} \mathrm{tr }\left((g\otimes \mathrm id _P)(\mathrm id _K \otimes f)\right) = \mathrm{tr }\left(\mathfrak s (f\otimes \mathrm id _L) (\mathrm id _Q\otimes g)\mathfrak s \right). \end{aligned}$$
(2.2)
In particular, for \(f:M\rightarrow N\) and \(g:N\rightarrow M\), we have \(\mathrm{tr }(g f) = \mathrm{tr }(f g)\).
This property can be proven directly with a long sequence of equations, but a much more conceptual proof is possible using string diagrams. In Fig. 5 we have drawn both sides of Eq. (2.2) as string diagrams. To prove Proposition 2.6 it then suffices to observe that one of these diagrams can be deformed into the other (this is easiest to see at first when \(Q=P=K=L=I\) are the unit object). The fundamental theorem of Joyal and Street [15] then implies that Eq. (2.2) holds in any symmetric monoidal category.
Fig. 5

Cyclicity of the symmetric monoidal trace

The symmetric monoidal trace satisfies many other useful naturality properties, most of which are likewise easiest to prove using string diagrams; see for instance [8, 17, 20, 21]. We will consider generalizations of many of these properties in Sect. 7.

Finally, as mentioned in the introduction, one of the main advantages of having an abstract formulation of trace is that disparate notions of trace which all fall into the general framework can be compared functorially. Recall that a lax symmetric monoidal functor \(F:\mathbf C \rightarrow \mathbf D \) between symmetric monoidal categories consists of a functor \(F\) and natural transformations
$$\begin{aligned} \mathfrak c :F(M)\otimes F(N)&\overset{}{\longrightarrow }&F(M\otimes N)\\ \mathfrak i :I_\mathbf{D }&\overset{}{\longrightarrow }&F(I_\mathbf{C }) \end{aligned}$$
satisfying appropriate coherence axioms. We say \(F\) is normal if \(\mathfrak i \) is an isomorphism, and strong if \(\mathfrak c \) and \(\mathfrak i \) are both isomorphisms.
When drawing string diagrams, we follow [23, 24] by imagining a monoidal functor as a kind of ‘fluid’ or ‘environment’ in which our strings and vertices can be immersed, and we notate such immersion graphically by a pattern of dots or lines characteristic of the functor. (We avoid the use of color, since that will be used to denote 0-cells when we come to string diagrams for bicategories in Sect. 3.) For simplicity, we continue to label strings and vertices by the objects and morphisms in the domain category \(\mathbf C \), since the presence of a functor pattern indicates application of the functor to yield corresponding objects and morphisms in \(\mathbf D \). Thus, for instance,
denotes the object \(M\!\in \!\mathbf C \), while
denotes the object \(F(M)\!\in \!\mathbf D \). With this notation, the structure maps \(\mathfrak c \) and \(\mathfrak i \) are shown in Fig. 6, and their coherence axioms in Fig. 7. (The last axiom is just the naturality of the transformation \(\mathfrak c \).) When either of these structure maps is an isomorphism, we draw its inverse in the same way, but upside down.
Fig. 6

The constraints of a lax monoidal functor

 
Fig. 7

The axioms of a lax symmetric monoidal functor

Unlike the situation for monoidal categories, it seems that string diagrams for monoidal functors have not yet been formalized. Thus, we must view their use as merely a convenient shorthand for writing out more precise proofs.

Proposition 2.7

Let \(F:\mathbf C \rightarrow \mathbf D \) be a normal lax symmetric monoidal functor, let \(M\in \mathbf C \) be dualizable with dual \({{M}^{\bigstar }}\), and assume that \(\mathfrak c :F(M)\otimes F({{M}^{\bigstar }})\rightarrow F(M\otimes {{M}^{\bigstar }})\) is an isomorphism.
  1. (i)

    Then \(F(M)\) is dualizable with dual \(F({{M}^{\bigstar }})\).

     
Also assume \(\mathfrak c :F(P) \otimes F(M) \rightarrow F(P\otimes M)\) is an isomorphism (as it is whenever \(P=I\), since \(F\) is normal).
  1. (ii)
    Then for any map \(f:Q\otimes M \rightarrow M\otimes P\), we have
    $$\begin{aligned} F(\mathrm{tr }(f)) = \mathrm{tr }\left(\mathfrak c ^{-1} \circ F(f)\circ \mathfrak c \right). \end{aligned}$$
    (2.3)
     
     
 
The evaluation and coevaluation of \(F(M)\) are shown graphically in Fig. 8. Note the need for invertibility of \(\mathfrak i \) and of the one relevant component of \(\mathfrak c \).
Fig. 8

Preservation of duals

The Eq. (2.3) is shown graphically in Fig. 9. We leave it to the reader to prove this equation graphically, using the coherence laws drawn in Fig. 7 along with naturality (moving unrelated morphisms past each other side-by-side) and the invertibility of \(\mathfrak i \) and \(\mathfrak c \).
Fig. 9

Preservation of traces

 

Example 2.8

As mentioned in Sect. 1, Proposition 2.7 implies the Lefschetz fixed point theorem, by the following argument. The rational chain complex functor is a strong symmetric monoidal functor from the stable homotopy category to the derived category of \(\mathbb Q \). Composing this functor with the homology functor, which is strong symmetric monoidal by the Künneth theorem, gives a strong symmetric monoidal functor to graded vector spaces. Applying the proposition to this composite of functors identifies the Lefschetz number with the fixed point index.

 

Example 2.9

A simpler example is given by extension of scalars. If \(R\) and \(S\) are commutative rings and \(\psi :R\rightarrow S\) is a ring homomorphism, extension of scalars is a strong symmetric monodal functor from \(R\)-modules to \(S\)-modules. If \(f:Q\otimes M\rightarrow M\otimes P\) is a map of \(R\)-modules, this proposition implies \(\mathrm{tr }(f\otimes _RS)=\mathrm{tr }(f)\otimes _RS\).

In addition to this “functoriality,” the symmetric monoidal trace also satisfies a sort of “2-functoriality.” Recall that if \(F,G:\mathbf C \rightarrow \mathbf D \) are lax symmetric monoidal functors, a monoidal natural transformation is a natural transformation \(\alpha :F\rightarrow G\) which is compatible with the monoidal constraints of \(F\) and \(G\) in an evident way. Graphically, we draw a monoidal natural transformation as a ‘membrane’ or ‘interface’ between regions denoting the two functors, as shown in Fig. 10a. Note that the morphism \(\alpha _{M}:F(M) \rightarrow G(M)\) is not explicitly pictured as a node, but rather implied as the \(M\)-string passes through the \(\alpha \) membrane.
Fig. 10

A monoidal natural transformation and its axioms

 

Proposition 2.10

Let \(F,G:\mathbf C \rightarrow \mathbf D \) be normal lax symmetric monoidal functors, let \(\alpha :F\rightarrow G\) be a monoidal natural transformation, let \(M\) be dualizable in \(\mathbf C \), and assume that \(F\) and \(G\) satisfy the hypotheses of Proposition 2.7. Then
  1. (i)

    \(\alpha _M:F(M)\rightarrow G(M)\) is an isomorphism, whose inverse is the mate of \(\alpha _{{{M}^{\bigstar }}}\), and

     
  2. (ii)
    For any \(f:Q\otimes M \rightarrow M\otimes P\), the following square commutes.
     
  3. (iii)

    In particular, for \(f:M\rightarrow M\), we have \(\mathrm{tr }(F(f)) = \mathrm{tr }(G(f))\).

     
 
Proposition 2.10(i) is known colloquially as “duals invert.” Half of it is pictured in Fig. 11a; it follows from the axioms in Fig. 10 and a triangle identity. The other half is of course similar. (This is one of the examples of string diagram calculus for monoidal functors given in [23].)
Fig. 11

2-functoriality of traces

A graphical picture of Proposition 2.10(ii) is shown in Fig. 11b. It is easy to give a direct proof of this equality by successive applications of the axioms in Fig. 10. In fact, there is an even easier proof: application of Proposition 2.7 to \(F\) and \(G\) reduces the desired statement to naturality of \(\alpha \). However, this easy proof will no longer be available in the bicategorical case.

Finally, Proposition 2.10(iii) follows because when \(F\) and \(G\) are normal, the component \(\alpha _{I}\) at the unit object is always an isomorphism.

Example 2.11

Extension of scalars along the inclusion \(\iota :\mathbb Z \rightarrow \mathbb Q \) defines a natural transformation from the functor \((H_*(-;\mathbb Z )/\mathrm{Torsion })\otimes \mathbb Q \) to the functor \(H_*(-;\mathbb Q )\). (We quotient \(H_*(-;\mathbb Z )\) by torsion in order to make it strong symmetric monoidal.) Proposition 2.10 then implies the familiar fact that the Lefschetz number computed using \(H_*(-;\mathbb Z )/\mathrm{Torsion }\) is the same as that computed using \(H_*(-;\mathbb Q )\). An analogous argument works for Lefschetz numbers computed at the level of chain complexes, in which case quotienting by torsion is no longer necessary since the chain complexes consist of free abelian groups.

 

3 Bicategories

As suggested in the introduction, sometimes we also need a notion of trace in non-commutative situations, where we don’t even have a monoidal category, let alone a symmetric one. For example, if \(R\) is a non-commutative ring, then there is no monoidal structure on the category of \(R\)-modules. What there is, however, is a tensor product of a right \(R\)-module with a left \(R\)-module, which is a special case of the tensor product of bimodules. An appropriate categorical context for this is a bicategory, originally due to [3].

Definition 3.1

A bicategory \(\fancyscript{B}\) consists of the following.

  • A collection of objects or 0-cells \(R,S,T,\ldots \).

  • For each pair of objects, a category \(\fancyscript{B}(R,S)\).

  • For each object, a unit \(U_R\in \fancyscript{B}(R,R)\).

  • For each triple of objects, a composition functor
    $$\begin{aligned} \odot :\fancyscript{B}(R,S)\times \fancyscript{B}(S,T)\overset{}{\longrightarrow }\fancyscript{B}(R,T). \end{aligned}$$
  • Natural isomorphisms
    $$\begin{aligned} \mathfrak a :M\odot (N\odot P)&\overset{\cong }{\longrightarrow } (M\odot N)\odot P\\ \mathfrak l :U_R \odot M&\overset{\cong }{\longrightarrow } M\\ \mathfrak r :M\odot U_S&\overset{\cong }{\longrightarrow } M \end{aligned}$$
    satisfying the same coherence axioms as for a monoidal category.
 

We call the objects of \(\fancyscript{B}(R,S)\) 1-cells and its morphisms 2-cells. We regard 1-cells and 2-cells in a bicategory as analogous to the objects and morphisms in a monoidal category, respectively, with the 0-cells playing a “bookkeeping” role informing us which 1-cells can be tensored with which others. We often write a 1-cell \(M\in \fancyscript{B}(R,S)\) as \(M:R \shortmid \!\!\!\rightarrow S\) to distinguish it from a 2-cell \(f:M\rightarrow N\).

Example 3.2

There is a bicategory \(\mathcal M od /\!_\mathcal{R ing }\) whose objects are not-necessarily-commutative rings, and such that \(\mathcal M od /\!_\mathcal{R ing }(R,S)\) is the category of \(R\)-\(S\)-bimodules. The unit \(U_R\) is \(R\) regarded as an \(R\)-\(R\)-bimodule, and the composition \(\odot \) is the tensor product of bimodules, \(M\odot N = M\otimes _S N\).

 

Example 3.3

If \(\mathbf C \) is a monoidal category, we have a bicategory \(\mathbf{C }/\!_\star \) with one object \(\star \) and \((\mathbf{C }/\!_\star )(\star ,\star )=\mathbf C \), \(U_\star = I\), and \(\odot =\otimes \). In this sense, one can think of a bicategory as “a monoidal category with many objects”.

 

Remark 3.4

Our naming convention for bicategories is a little unusual: to clarify the discussion of later examples, we have chosen to use names which indicate both the objects and the 1-cells. Thus, for example, the name \(\mathcal M od /\!_\mathcal{R ing }\), which we read as “modules over rings,” indicates that the objects are rings and the 1-cells are modules, while the name \(\mathbf{C }/\!_\star \) indicates that there is a unique object \(\star \) and the 1-cells are the objects of \(\mathbf C \).

 

Remark 3.5

We have chosen to write the composite of 1-cells \(M:R \shortmid \!\!\!\rightarrow S\) and \(N:S \shortmid \!\!\!\rightarrow T\) in a bicategory as \(M\odot N:R \shortmid \!\!\!\rightarrow T\), rather than \(N\odot M\). This is called diagrammatic order and makes sense for bicategories such as \(\mathcal M od /\!_\mathcal{R ing }\); if we had chosen the other order we would have to define \(\mathcal M od /\!_\mathcal{R ing }(R,S)\) to be the category of \(S\)-\(R\)-bimodules in order to have \(M\odot N = M\otimes _S N\).

There are other sorts of bicategories, however, for which the opposite choice makes more sense. For instance, there is a bicategory \(\mathcal C at \) whose objects are categories, whose 1-cells are functors, and whose 2-cells are natural transformations; in this case it makes more sense to write the composite of functors \(F:A\rightarrow B\) and \(G:B\rightarrow C\) as \(G \circ F:A\rightarrow C\).

Finally, string diagrams in a bicategory are also obtained by Poincaré duality, but one dimension up. Now 0-cells are represented by 2-dimensional regions (in color or shading, depending on whether the reader is fortunate enough to be reading this paper in color), 1-cells are represented by strings, and 2-cells are represented by vertices; see Fig. 12. Note that since 1-cells and 2-cells in a bicategory are analogous to objects and morphisms in a monoidal category, respectively, we can regard bicategorical string diagrams as obtained from those for monoidal categories by adding 0-cell labels to the 2-dimensional regions. However, the strings can no longer be crossed over each other, since there is no “symmetry” isomorphism in a bicategory.
Fig. 12

String diagrams for bicategories (color figure online)

 

4 Shadows

We will give further examples of bicategories shortly, but first we introduce the additional structure that we will need in order to define traces. Extra structure is necessary because in defining the symmetric monoidal trace, we used the symmetry isomorphism \(M\otimes {{M}^{\bigstar }} \cong {{M}^{\bigstar }}\otimes M\), whereas for 1-cells \(M:R \shortmid \!\!\!\rightarrow S\) and \(N:S \shortmid \!\!\!\rightarrow R\) in a bicategory, we cannot even ask whether \(M\odot N\) and \(N\odot M\) are isomorphic, since they are objects of different categories: one is a 1-cell \(R \shortmid \!\!\!\rightarrow R\) and the other is a 1-cell \(S \shortmid \!\!\!\rightarrow S\). For example, if \(R\) is a noncommutative ring, \(M\) is a left \(R\)-module, and \(N\) is a right \(R\)-module, then \(M\odot N\) is an \(R\)-\(R\)-bimodule, while \(N\odot M\) is just an abelian group.

In this latter example, there is a naive way to compare the two: if we quotient out the \(R\)-\(R\)-bimodule structure on \(M\odot N\), we obtain an abelian group, which is in fact isomorphic to \(N\odot M\). This approach turns out to be surprisingly effective, and moreover many bicategories admit a similar sort of “quotienting” operation, as we will see. Thus, we encapsulate the important properties of such an operation in the following abstract definition.

Definition 4.1

[29] Let \(\fancyscript{B}\) be a bicategory. A shadow functor for \(\fancyscript{B}\) consists of functors
$$\begin{aligned} \big \langle \big \langle - \big \rangle \big \rangle :\fancyscript{B}(R,R) \rightarrow \mathbf T \end{aligned}$$
for each object \(R\) of \(\fancyscript{B}\) and some fixed category \(\mathbf T \), equipped with a natural isomorphism
$$\begin{aligned} \theta :\langle \langle M\odot N \rangle \rangle \overset{\cong }{\longrightarrow } \langle \langle N\odot M \rangle \rangle \end{aligned}$$
for \(M:R \shortmid \!\!\!\rightarrow S\) and \(N:S \shortmid \!\!\!\rightarrow R\) such that the following diagrams commute whenever they make sense:

 

Remark 4.2

The above hexagon is not one of the hexagon axioms for a braided monoidal category. In fact, if \(\theta \) were viewed as a “braiding”, the two sides of this hexagon would describe unequal braids. The shadow axioms describe a “cyclic” operation, rather than a “linear” one that happens to have transpositions.

It may seem that we should require \(\theta ^2=1\), but in fact this is automatic:

Proposition 4.3

If \(\langle \langle - \rangle \rangle \) is a shadow functor on \(\fancyscript{B}\), then the composite
is the identity.

 

Proof

Let \(M:R \shortmid \!\!\!\rightarrow S\) and \(N:S \shortmid \!\!\!\rightarrow R\), and set \(P=U_R\) in the hexagon axiom for a shadow functor. Naturality of \(\theta \), the unit axioms for a shadow functor, and the axioms relating \(\mathfrak l ,\mathfrak r \) to \(\mathfrak a \) in a bicategory reduce the hexagon to the desired statement. \(\square \)

 

Remark 4.4

In the proof of Proposition 4.3 we didn’t use the assumption that \(\theta \) is an isomorphism. Thus, we could just as well have dispensed with that assumption in the definition. It is also worth noting that if we assume Proposition 4.3, then we can derive either of the unit axioms from the other.

 

Example 4.5

Let \(\mathbf C \) be a symmetric monoidal category; then its identity functor gives \(\mathbf{C }/\!_\star \) a canonical shadow functor with \(\mathbf T =\mathbf C \). The isomorphism \(\theta \) is induced by the symmetry of \(\mathbf C \). By Remark 4.2, it does not suffice for \(\mathbf C \) to be braided.

 

Example 4.6

We define the shadow of a bimodule \(M:R \shortmid \!\!\!\rightarrow R\) in \(\mathcal M od /\!_\mathcal{R ing }\) to be the coequalizer
$$\begin{aligned} R\otimes M \rightrightarrows M \rightarrow \langle \langle M \rangle \rangle , \end{aligned}$$
where the two parallel maps are the left and right actions of \(R\). Here \(\mathbf T \) is abelian groups and \(\langle \langle M \rangle \rangle \) is the abelian group obtained from \(M\) by forcing the left and right actions of \(R\) to be equal; it might be called an “underived version of Hochschild homology”. The isomorphism \(\theta \) is obvious.

We will consider more examples in Sect. 6 after defining traces.

Finally, given that a shadow is a fundamentally cyclic operation, it is natural to represent it by closing up planar bicategorical string diagrams into a cylinder, thereby allowing strings to migrate cyclically around the back of the cylinder to the other side. This is illustrated in Fig. 13. In Appendix  we will extend the fundamental theorem of Joyal and Street to bicategories with shadows, by defining the “value” of such a labeled cylindrical diagram and proving that it is invariant under deformation.
Fig. 13

String diagrams for shadows (color figure online)

 

Remark 4.7

The boundary circles of these cylinders should not be regarded as fixed, but are free to rotate as we deform the diagram. For instance, this means that the two pictures shown in Fig.  14 should be regarded as the same. (Technically, we do have to distinguish a “basepoint” or “cut point” on the top and bottom circles in order to assign a well-specified value, since \(\langle \langle M\odot N \rangle \rangle \) is rarely equal to \(\langle \langle N\odot M \rangle \rangle \). These basepoints then have to rotate along with the corresponding boundary circle. This will be made precise in Appendix . When drawing pictures in the rest of the paper, we will always assume that these basepoints are in the back.)

 
Fig. 14

Cylinder boundaries are not fixed (color figure online)

 

5 Duality and trace

We say that a 1-cell \(M:R \shortmid \!\!\!\rightarrow S\) in a bicategory is right dualizable if there is a 1-cell \({{M}^{\bigstar }}:S \shortmid \!\!\!\rightarrow R\), called its right dual, and evaluation and coevaluation 2-cells \(\eta :U_R \rightarrow M\odot {{M}^{\bigstar }}\) and \(\varepsilon :{{M}^{\bigstar }}\odot M\rightarrow U_S\) satisfying the triangle identities. We then also say that \((M,{{M}^{\bigstar }})\) is a dual pair, that \({{M}^{\bigstar }}\) is left dualizable, and that \(M\) is its left dual. See Fig. 15 for the string diagrams for coevaluation and evaluation.
Fig. 15

Coevaluation and evaluation in a bicategory (color figure online)

In the bicategorical context, duals are also frequently called adjoints, since in the bicategory \(\mathcal C at \) (see Remark 3.5) a dual pair is just an adjoint pair of functors. As in the monoidal case, there are equivalent characterizations of duals, especially when the bicategory \(\fancyscript{B}\) is closed; see, for instance, [22, 16.4]. Note that the notion is now asymmetric: right duals and left duals are different.

When \(M\) and \(N\) are right dualizable, every 2-cell \(f:Q\odot M\rightarrow N\odot P\) has a mate \({{f}^{\bigstar }}:{{N}^{\bigstar }}\odot Q\rightarrow P\odot {{M}^{\bigstar }}\), defined as the composite
$$\begin{aligned} {{N}^{\bigstar }}\!\odot \!Q \xrightarrow {1\odot \eta } {{N}^{\bigstar }}\!\odot \!Q\odot M \odot {{M}^{\bigstar }} \xrightarrow {1\odot f \odot 1} {{N}^{\bigstar }}\odot N \odot P \!\odot \!{{M}^{\bigstar }} \xrightarrow {\varepsilon \odot 1} P\odot {{M}^{\bigstar }}. \end{aligned}$$
This operation is pictured graphically in Fig. 16a.
Fig. 16

Mates in a bicategory (color figure online)

There is an evident dual construction for left dualizable objects, and the triangle identities show that the two are inverses. Therefore, giving a 2-cell \(Q\odot M\rightarrow N\odot P\) is equivalent to giving a 2-cell \({{N}^{\bigstar }}\odot Q\rightarrow P\odot {{M}^{\bigstar }}\). For this reason, it is justified to draw horizontal strings labeled by dual pairs, as in Fig. 16b, since they can be “tipped” up or down to represent a 2-cell or its mate, as needed. (We could also consider such pictures as taking place in a double category whose vertical arrows are dual pairs, such as is used in [19] to describe naturality properties of mates.)

Definition 5.1

Let \(\fancyscript{B}\) be a bicategory with a shadow functor and \(M\) a dualizable 1-cell of \(\fancyscript{B}\). The trace of a 2-cell \(f:Q\odot M\rightarrow M\odot P\) is the composite:
$$\begin{aligned} \ \langle \langle Q \rangle \rangle \xrightarrow {\langle \langle \mathrm id \odot \eta \rangle \rangle } \langle \langle Q\!\odot \! M\!\odot \! {{M}^{\bigstar }} \rangle \rangle \xrightarrow {\langle \langle f\odot \mathrm id \rangle \rangle } \langle \langle M\!\odot \! P\!\odot \! {{M}^{\bigstar }} \rangle \rangle \overset{\theta }{\longrightarrow } \langle \langle {{M}^{\bigstar }}\!\odot \! M\!\odot \! P \rangle \rangle \xrightarrow {\langle \langle \varepsilon \odot \mathrm id \rangle \rangle } \langle \langle P \rangle \rangle . \end{aligned}$$
 
The trace is an arrow from \(\langle \langle Q \rangle \rangle \) to \(\langle \langle P \rangle \rangle \) in the target category \(\mathbf T \) for the shadow, and is independent of the choice of \({{M}^{\bigstar }}, \eta ,\) and \(\varepsilon \). A similar, but not identical, definition works when \(M\) is left dualizable instead. A string diagram picture of this trace is shown in Fig. 17, along with an equivalent, more concise version using the above convention for drawing dual-pair strings horizontally.
Fig. 17

The bicategorical trace (color figure online)

Of course, if we take \(Q=U_{R}\) and \(P=U_{S}\), we obtain the trace of an endomorphism. The frequency with which this case occurs implies that the shadows of unit 1-cells are particularly important. We refer to the shadow of \(U_R\) as “the shadow of \(R\)” and write \(\langle \langle R \rangle \rangle =\langle \langle U_R \rangle \rangle \).

Example 5.2

Of course, in \(\mathbf{C }/\!_\star \) the bicategorical trace reduces to the canonical symmetric monoidal trace.

 

Example 5.3

In \(\mathcal M od /\!_\mathcal{R ing }\), a bimodule \(M:\mathbb Z \shortmid \!\!\!\rightarrow R\) is right dualizable when it is a finitely generated projective (right) \(R\)-module. The trace of an endomorphism \(f:M\rightarrow M\) is then a map from \(\langle \langle \mathbb Z \rangle \rangle =\mathbb Z \) to \(\langle \langle R \rangle \rangle \), which is determined by the image of \(1\). This element of \(\langle \langle R \rangle \rangle \) is known as the Hattori-Stallings trace of \(f\), see [11, 37].

More generally, if \(\psi :R\rightarrow R\) is a ring homomorphism and \(f:M\rightarrow M\) is a \(\psi \)-equivariant map (meaning that \(f(mr) = f(m)\psi (r)\)), we can view \(f\) as a 2-cell \(M\rightarrow M\odot R_\psi \) in \(\mathcal M od /\!_\mathcal{R ing }\). Here \(R_\psi \) denotes \(R\) regarded as an \(R\)-\(R\)-bimodule with the right action twisted by \(\psi .\) The trace of \(f\) is then a map \(\mathbb Z \cong \langle \langle \mathbb Z \rangle \rangle \rightarrow \langle \langle R_\psi \rangle \rangle \) in \(\mathbf Ab \), or equivalently an element of \(\langle \langle R_\psi \rangle \rangle \). (Explicitly, \(\langle \langle R_\psi \rangle \rangle \) is the abelian group obtained from \(R\) by quotienting by \(r s\sim s \psi (r)\) for all \(r\).)

Note that even when \(R\) is commutative, this latter type of trace cannot be expressed without bicategorical technology, since \(R_\psi \) is unavoidably an \(R\)-\(R\)-bimodule, and the category of \(R\)-\(R\)-bimodules is not symmetric (or even braided) monoidal.

 

Remark 5.4

The categorically inclined reader will observe that a shadow is a “categorified trace”. Just as trace is a cyclic function on endomorphisms in a 1-category, shadow is a cyclic functor on endo-1-cells in a bicategory. Thus, to define traces in a bicategory, the bicategory must be equipped with a categorified trace, just as (for example) to define monoids in a category, the category must be equipped with a monoidal structure. This is a version of the microcosm principle of [2].

Since there is a canonical (in fact, unique) trace in any symmetric monoidal category in which all objects are dualizable, it is natural to think of constructing a canonical shadow in an analogous way for any symmetric monoidal bicategory with duals for objects (see [5]). In fact, all of our examples of shadows on bicategories can be constructed in this way, and our cylindrical string diagrams can thereby be identified with a fragment of the expected “surface diagrams” that apply to monoidal bicategories (see [39]). However, in applications we prefer to avoid this abstract point of view, since a symmetric monoidal bicategory with duals for objects is quite a complicated object, while the shadows arising in practice have much simpler descriptions.

 

6 Examples of shadows and traces

In this section we will consider several different examples of bicategorical shadows and traces. We begin with a generalization of Example 5.3.

Example 6.1

In Example 5.3 we considered right \(R\)-modules as 1-cells \(\mathbb Z \rightarrow R\) in \(\mathcal M od /\!_\mathcal{R ing }\). More generally, an \(S\)-\(R\)-bimodule \(M\) is right dualizable in \(\mathcal M od /\!_\mathcal{R ing }\) if and only if it is finitely generated and projective as a right \(R\)-module. The Euler characteristic of such an \(M\) (that is, the trace of its identity map) is the map \(\langle \langle S \rangle \rangle \rightarrow \langle \langle R \rangle \rangle \) which sends each \(s\in S\) to the Hattori-Stallings trace of the \(R\)-module homomorphism \((s\cdot -):M\rightarrow M\) (which descends to \(\langle \langle S \rangle \rangle \) because the Hattori-Stallings trace is cyclic). More generally, the trace of \(f:M\rightarrow M\) is the map \(\langle \langle S \rangle \rangle \rightarrow \langle \langle R \rangle \rangle \) determined by sending \(s\in S\) to the Hattori-Stallings trace of \(m\mapsto f(sm)=s\cdot f(m)\).

In particular, let \(G\) be a group, \(k\) a field, and \(V\) a finite-dimensional left \(k G\)-module (that is, a representation of \(G\)). If we regard \(V\) as a 1-cell \(k G\rightarrow k\) in \(\mathcal M od /\!_\mathcal{R ing }\), then it is right dualizable, so it has an Euler characteristic, which is a map \(\langle \langle k G \rangle \rangle \rightarrow k\). It is easy to check that \(\langle \langle k G \rangle \rangle \) is the \(k\)-vector space generated by the conjugacy classes of \(G\), and this Euler characteristic is essentially the character of the representation \(V\).

The following two examples are somewhat degenerate, but can be useful.

Example 6.2

Let \(\mathbf S \) be a category with pullbacks (a.k.a. fiber products) and define a bicategory \(\mathbf S /\text{ s}\) whose objects are those of \(\mathbf S \) and whose 1-cells \(R\rightarrow S\) are diagrams \(R\leftarrow M\rightarrow S\) in \(\mathbf S \); these are often called spans or correspondences. Composition is by pullback. We define the shadow of a span \(R\leftarrow M\rightarrow R\) to be the object \(P\) in the pullback square

This defines a shadow on \(\mathbf S /\text{ s}\) with values in \(\mathbf S \). If \(f:R\leftarrow R\) is an arrow in \(\mathbf S \), the shadow of the corresponding span \(R\xleftarrow {{{\ \text{ id}\ }}} R \xrightarrow {\ f\ }R\) (the “graph” of \(f\)) is the equalizer of \(f\) and \(\mathrm id _R\); that is, the “object of fixed points” of \(f\). In particular, for an object \(R\) of \(\mathbf S \), we have \(\langle \langle R \rangle \rangle = R\).

In \(\mathbf S /\text{ s}\), a 1-cell \(R\!\xleftarrow {{{\ {h}\ }}}\! M\!\xrightarrow {\ g\ }\! S\) is right dualizable if and only if \(h\) is an isomorphism. If \(R\xleftarrow {{{\ \text{ id}\ }}} R\xrightarrow {\ g\ } S\) is such a dualizable 1-cell, an endomorphism of it is a map \(f:R\rightarrow R\) such that \(gf=g\). Its trace is precisely \(gf=g:\langle \langle R \rangle \rangle = R \rightarrow S = \langle \langle S \rangle \rangle \).

 

Example 6.3

There is a bicategory \(n\mathcal C ob /\!_\mathcal{M fd }\) whose objects are closed \((n-1)\)-dimensional manifolds, whose 1-cells are \(n\)-dimensional cobordisms, and whose 2-cells are boundary-preserving diffeomorphisms. To make the associativity and unit isomorphisms coherent, we have to include “thin” cobordisms, and also give “collars” to the thick ones. A cobordism from \(R\) to \(R\) is an \(n\)-manifold \(M\) whose boundary is \(R\sqcup R^{{op}}\); we define its shadow \(\langle \langle M \rangle \rangle \) to be the result of gluing together these two copies of \(R\).

There are no interesting bicategorical traces in \(n\mathcal C ob /\!_\mathcal{M fd }\), since all the 2-cells are isomorphisms. (In essence, here the shadow itself “is” the interesting trace.)

Many bicategories of interest are obtained as “homotopy bicategories” of other bicategories, by which we mean that we invert some class of 2-cells considered as “weak equivalences.” The theory of homotopy bicategories is not well-developed, but since our goal is only to give examples, we will gloss over all the technicalities.

Example 6.4

Let \(\mathcal C h /\!_\mathcal{R ing }\) be the bicategory whose objects are rings (not necessarily commutative) and whose 1-cells are chain complexes of bimodules. We define shadows analogously to \(\mathcal M od /\!_\mathcal{R ing }\), introducing appropriate signs in the definition of \(\theta \). Here \(\mathbf T \) is the category \(\mathbf Ch _\mathbb{Z }\) of chain complexes of abelian groups.

We can now construct a bicategory \(\mathrm Ho (\mathcal C h /\!_\mathcal{R ing })\), whose objects are the same as those as \(\mathcal C h /\!_\mathcal{R ing }\), but in which \(\mathrm Ho (\mathcal C h /\!_\mathcal{R ing })(A,B)\) is the derived category of \(A\otimes B^{ op }\) (that is, we invert the quasi-isomorphisms of chain complexes). The composition \(\odot \) is now the derived functor of the tensor product of bimodules, and the shadow of a 1-cell \(M:A\rightarrow A\) now is the Hochschild homology \(\mathop {H\!H}(A;M)\) of \(A\) with coefficients in \(M\). The target category \(\mathbf T \) of the shadow is now \(\mathrm Ho (\mathbf Ch _\mathbb{Z })\).

Similarly, we have a bicategory \(\mathcal C h /\!_{ DGA }\)whose objects are DGAs rather than rings, and \(\mathrm Ho (\mathcal C h /\!_{ DGA })\) is defined similarly. For technical reasons, we may sometimes want to restrict the objects of \(\mathrm Ho (\mathcal C h /\!_{ DGA })\) to be DGAs which are cofibrant, at least as chain complexes if not in some model structure for DGAs. We can also consider modules over ring spectra, in which case the shadow is topological Hochschild homology.

 

Example 6.5

There is a bicategory \(\mathcal T \! op _*/\!_\mathcal{G rp }\) whose objects are discrete groups, whose 1-cells \(G\rightarrow H\) are based topological spaces with a left action of \(G\) and a right action of \(H\), and whose 2-cells are equivariant maps. The unit \(U_G\) is \(G_+\) regarded as a \(G\)-\(G\) space, and the composition of \(M:G\rightarrow H\) and \(N:H\rightarrow K\) is the smash product over \(H\), i.e. the coequalizer
$$\begin{aligned} M\wedge H_+\wedge N\rightrightarrows M\wedge N\rightarrow M\odot N. \end{aligned}$$
Likewise, the shadow of a \(G\)-\(G\) space \(M\) is the coequalizer
$$\begin{aligned} G_+\wedge M\rightrightarrows M\rightarrow \langle \langle M \rangle \rangle \end{aligned}$$
where the two parallel maps are the left and right actions of \(G\) on \(M\). The target category \(\mathbf T \) of the shadow is the category \(\mathbf Top _*\) of based spaces.
For a useful homotopy version of this example, we need to “stabilize” (i.e. move from spaces to spectra, as in Example 2.4) as well as pass to homotopy categories. This would result in a bicategory we refer to as \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\), whose shadow would take values in the ordinary stable homotopy category \(\mathrm Ho (\mathbf Sp )\). A careful foundational analysis leading to such a bicategory has not yet been done, but we can still use a naive sort of stabilization to obtain a notion of duality analogous to Spanier-Whitehead duality. That is, we say a 1-cell \(M\) in \(\mathcal T \! op _*/\!_\mathcal{G rp }\) is \(n\)-dualizable if we have maps
$$\begin{aligned} G_+\wedge S^n\rightarrow M\odot N N\odot M\rightarrow H_+\wedge S^n \end{aligned}$$
satisfying the usual relations up to equivariant stable homotopy. Once \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\) is shown to exist, we expect its intrinsic notion of duality to be describable as \(n\)-duality; thus from now on we will refer informally to “duality” and “trace” in \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\).

It turns out that very little is known about this sort of duality, but there are a couple examples that have interesting traces. On the one hand, if \(G\) is finite and \(M\) has a free \(G\)-action, then duality for \(M\), regarded as a 1-cell \(1\rightarrow G\) in \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\), is equivalent to duality for \(M\) in \(\mathrm Ho ({G}\text{-}\mathbf Sp )\), the equivariant stable homotopy category for \(G\); see [1, 8.6]. The corresponding traces in \(\mathrm Ho ({G}\text{-}\mathbf Sp )\) and in \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\) are not identical, but the first is a direct summand of the second. Both have interpretations as equivariant fixed point indices, but the trace in \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\) also detects fixed orbits; see [28].

On the other hand, if \(G\) is discrete (possibly infinite) and \(M\) is a finite free right \(G\)-CW complex, then \(M:1\rightarrow G\) is dualizable in \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\). This is known as Ranicki duality; see [33]. For example, if \(X\) is a closed smooth manifold (or a compact ENR), its universal cover \(\tilde{X}\) has a free action of \(\pi _1(X)\) and this equivariant space is Ranicki dualizable.

Now suppose that \(X\) is a space of this sort, and let \(f:X\rightarrow X\) be a continuous endomorphism. For each choice of base point \(*\) in \(X\) and base path from \(*\) to \(f(*)\), there is an induced group homomorphism
$$\begin{aligned} \psi :\pi _1(X)\rightarrow \pi _1(X) \end{aligned}$$
and a \(\psi \)-equivariant map
$$\begin{aligned} \tilde{f}:\tilde{X}\rightarrow \tilde{X}. \end{aligned}$$
As in Example 5.3, we can regard \(\tilde{f}\) as a 2-cell \(\tilde{X} \rightarrow \tilde{X} \odot \pi _1(X)_\psi \), where \(\pi _1(X)_\psi \) denotes \(\pi _1(X)\) regarded as a \(\pi _1(X)\)-\(\pi _1(X)\)-space with the right action twisted by \(\psi \). The trace of \(\tilde{f}\) is, by definition, an element of the zeroth stable homotopy group of the discrete set \(\langle \langle \pi _1(X)_\psi \rangle \rangle \), which is just the free abelian group \(\mathbb Z \langle \langle \pi _1(X)_\psi \rangle \rangle \).

This trace is known as the Reidemeister trace. It is a refinement of the fixed-point index, which separates out contributions coming from different “fixed-point classes”. A classical description of the Reidemester trace can be found in [4, 12]; the bicategorical description is due to [29]. In Sect. 8, we will see that the abstract proof of the Lefschetz fixed-point theorem from Example 2.8 carries over directly to the Reidemeister trace when expressed bicategorically. In fact, the Reidemester trace is refined enough to support a converse to the Lefschetz fixed point theorem, but that requires a deeper, more concrete, argument.

Example 6.6

The authors of [22] define a bicategory whose objects are topological spaces, and in which a 1-cell \(R\shortmid \!\!\!\rightarrow S\) is a parametrized spectrum over \(R\times S\). We call this bicategory \(\mathrm Ho (\mathcal S p /\!_\mathcal{T op })\), since it is also the homotopy bicategory of a point-set-level bicategory \(\mathcal S p /\!_\mathcal{T op }\). If \(M:R\shortmid \!\!\!\rightarrow R\) is a spectrum parametrized over \(R\times R\), we define its shadow to be \(\langle \langle M \rangle \rangle = r_!\Delta ^*M\), where \(\Delta ^*\) denotes pullback along the diagonal \(R\rightarrow R\times R\) and \(r_!\) denotes pushforward along the map \(R\rightarrow \star \) to the point. This defines a shadow on \(\mathrm Ho (\mathcal S p /\!_\mathcal{T op })\) landing in the ordinary stable homotopy category \(\mathrm Ho (\mathbf Sp )\).

Using the identifications \( (B\times \star )\cong B\cong (\star \times B)\), a spectrum \(M\) over \(B\) can also be regarded as a 1-cell \(\hat{M}:B \shortmid \!\!\!\rightarrow \star \) or \(\check{M}:\star \shortmid \!\!\!\rightarrow B\) in \(\mathrm Ho (\mathcal S p /\!_\mathcal{T op })\). The 1-cell \(\hat{M}\) is right dualizable if and only if \(M\) is dualizable in the symmetric monoidal category \(\mathrm Ho (\mathbf Sp _{B})\) of parametrized spectra over \(B\). In this case \(M\) is said to be fiberwise dualizable, since the fiberwise suspension spectrum of a fibration is dualizable in this sense if and only if each fiber is dualizable in \(\mathrm Ho (\mathbf Sp )\); see [22, 15.1.1].

Right dualizability of \(\check{M}\), called Costenoble-Waner duality, is very different. For example, if the fiberwise sphere spectrum \(S_B= \Sigma ^{\infty }_{B,+}(B)\) is Costenoble-Waner dualizable, then \(\Sigma ^\infty _+(B)\) is necessarily dualizable in \(\mathrm Ho (\mathbf Sp )\); see [22, 18.1.6]. In other words, fiberwise duality contains information only about the fibers of a parametrized space, while Costenoble-Waner duality also knows about the base space.

The traces of endomorphisms of \(M\), \(\hat{M}\), and \(\check{M}\) are also closely related. We will study these relationships in a more general setting in [31]; here we mention only the simplest case of Euler characteristics. If \(M\) is fiberwise dualizable, then the Euler characteristics of \(M\) (in \(\mathrm Ho (\mathbf Sp _B)\)) and \(\hat{M}\) (in \(\mathrm Ho (\mathcal S p /\!_\mathcal{T op })\)) are defined, and the following triangle commutes:
where \(S_B\) is the parametrized sphere spectrum over \(B\), \(S\) is the ordinary sphere spectrum, and the map \(r_! S_B \rightarrow S\) is adjunct to the defining isomorphism \(S_B \cong r^* S\). On the other hand, if \(M\) is Costenoble-Waner dualizable, then \(r_!M\) is dualizable, so that the Euler characteristics of \(\check{M}\) (in \(\mathrm Ho (\mathcal S p /\!_\mathcal{T op })\)) and \(r_!M\) (in \(\mathrm Ho (\mathbf Sp )\)) are defined, and the following triangle commutes:

Note that in the first case, the Euler characteristic of \(\hat{M}\) contains less information than that of \(M\), while in the second case, the Euler characteristic of \(\check{M}\) contains more information than that of \(r_!M\).

Both kinds of dualizable parametrized spectra often arise as fiberwise suspension spectra of parametrized spaces, i.e. by applying a functor from \(\text{ Top/}_{\text{ Top}}\) to \(\mathcal S p /\!_\mathcal{T op }\). Thus, this is a bicategorical instance of Remark 2.5, since \(\text{ Top/}_{\text{ Top}}\) is “cartesian” and has no interesting dual pairs itself.

Like Example 6.5, traces of some parameterized endomorphisms have familiar fixed-point interpretations. If \(M\) is a parameterized space over \(B\) whose suspension spectrum is fiberwise dualizable and \(f:M\rightarrow M\) is a fiberwise endomorphism, the trace of the suspension of \(f\) is the fiberwise fixed point index of \(f\), [7]. If \(B\) is a closed smooth manifold (or compact ENR), \(S_B^0\) is Costenoble-Waner dualizable. An endomorphism \(f\) of \(B\) induces a “twisted endomorphism” of \(S_B^0\) and the trace of the twisted endomorphism is the Reidemeister trace of \(f\); see [27].

 

Example 6.7

A simpler bicategorical instance of Remark 2.5 is provided by bicategories of matrices. Let \(\mathcal M at (\mathbf Ab )/\!_\mathcal{S et }\) denote the bicategory whose objects are sets, whose 1-cells \(R\shortmid \!\!\!\rightarrow S\) are \((R\times S)\)-matrices \((M_{r,s})_{r\in R,s\in S}\) of abelian groups, and whose composition \(\odot \) is given by “matrix multiplication:”
$$\begin{aligned} \left(M\odot N\right)_{r,t} = \bigoplus _{s\in S} \left(M_{r,s} \otimes N_{s,t}\right). \end{aligned}$$
The shadow of a square matrix \((M_{r_1,r_2})_{r_1,r_2\in R}\) is given by its “trace:” \(\langle \langle M \rangle \rangle = \bigoplus _{r\in R} M_{r,r}\). The unit \(U_R\) is given by
$$\begin{aligned} (U_R)_{r_1,r_2} = \left\{ \begin{array}{l@{\quad }l} \mathbb Z&\text{ if} r_1=r_2\\ 0&\text{ otherwise} \end{array} \right. \end{aligned}$$
and so the shadow of a set \(R\) is \(\langle \langle R \rangle \rangle = \bigoplus _R \mathbb Z = \mathbb Z [R]\).
It is easy to construct a functor of bicategories \(\mathbb Z [-]:\text{ Set/}_{\text{ Set}}\rightarrow \mathcal M at (\mathbf Ab )/\!_\mathcal{S et }\) which is the identity on 0-cells, and which turns a span \(R\leftarrow M \rightarrow S\) into a matrix whose \((r,s)\)-entry is the free abelian group on the fiber of \(M\) over \((r,s)\). Moreover, if \(S\) is finite, and so is each such fiber, then \(\mathbb Z [M]:R\shortmid \!\!\!\rightarrow S\) is dualizable in \(\mathcal M at (\mathbf Ab )/\!_\mathcal{S et }\). (The case \(S=\star \) is an analogue of fiberwise duality, while the case \(R=\star \) is an analogue of Costenoble-Waner duality; in [31] we will unify these examples in a general context.) Finally, if \(f:M\rightarrow M\) is an endo-2-cell in \(\text{ Set/}_{\text{ Set}}\), then
$$\begin{aligned} \mathrm{tr }(\mathbb Z [f]):\mathbb Z [R]\rightarrow \mathbb Z [S] \end{aligned}$$
is the function which maps each generator \(r\in R\) to the sum \(\sum _{s\in S} \mathrm{ind }(f_{r,s})\cdot s\), where \(\mathrm{ind }(f_{r,s})\) is the number of fixed points of \(f_{r,s}:M_{r,s}\rightarrow M_{r,s}\).

There are a number of other examples, some of which can be found in [28, 29, 30], but most can be considered extensions of those we have mentioned above. One other large class of examples consists of “monoids and bimodules” in some other bicategory; see for instance [29, 9.4] and [36, Sect. 11]. And as mentioned above, in [31] we will study a general class of examples including both \(\mathcal S p /\!_\mathcal{T op }\) and \(\mathcal M at (\mathbf Ab )/\!_\mathcal{S et }\).

7 Properties of trace in bicategories with shadows

We collect here some basic properties of the bicategorical trace, most of them analogous to the well-known properties of the symmetric monoidal trace. Like the latter, they are easiest to prove using string diagram calculus. In this section, we assume that \(\fancyscript{B}\) is a bicategory equipped with a shadow functor.

Proposition 7.1

[“Tightening”] Let \(M\) be right dualizable, let \(f:Q\otimes M \rightarrow M\otimes P\) be a 2-cell, and let \(g:Q^{\prime }\rightarrow Q\) and \(h:P\rightarrow P^{\prime }\) be 2-cells. Then
$$\begin{aligned} \langle \langle h \rangle \rangle \circ \mathrm{tr }(f)\circ \langle \langle g \rangle \rangle = \mathrm{tr }\big ((\mathrm id _M\otimes h)\circ f \circ (g\otimes \mathrm id _M)\big ). \end{aligned}$$
 
The above equality is shown graphically in Fig. 18. It should be easy to visualize a deformation relating the two pictures: we simply “pull on the string” through \(g\), \(f\), and \(h\) (hence the name, which we have taken from [17]).
Fig. 18

String diagram picture of Proposition 7.1 (tightening) (color figure online)

 

Proposition 7.2

[“Sliding” or “Cyclicity”] Let \(M\) and \(N\) be right dualizable 1-cells in \(\fancyscript{B}\) and \(g:K\odot N \rightarrow M\odot L\) and \(f:Q\odot M\rightarrow N\odot P\) be 2-cells. Then the following square commutes:
This equality is shown graphically in Fig. 19. Here the deformation requires a little more imagination: the idea is to slide \(f\) to the right and around the back of the cylinder, keeping the strings labeled \(Q\), \(K\), \(L\), and \(P\) fixed where they hit the top and bottom boundaries of the cylinder.
Fig. 19

String diagram picture of Proposition 7.2 (sliding) (color figure online)

 

Corollary 7.3

If \(M\) and \(N\) are right dualizable and \(f:M\rightarrow N\) and \(g:N\rightarrow M\) are 2-cells, then \(\mathrm{tr }(f g) = \mathrm{tr }(g f)\).

Of course, the unit \(U_R\) is always its own dual.

Proposition 7.4

If \(f:Q\odot U_R \rightarrow U_R\odot P\) is any 2-cell, we have \(\mathrm{tr }(f) = \langle \langle f \rangle \rangle \)  .

The string diagram picture of Proposition 7.4 is so tautologous as to not be worth drawing, since unit 1-cells are represented by empty space.

If \(M\) and \(N\) are right dualizable with right duals \({{M}^{\bigstar }}\) and \({{N}^{\bigstar }}\), then \(M\odot N\) is right dualizable with right dual \({{N}^{\bigstar }}\odot {{M}^{\bigstar }}\). (This can be a source of many dual pairs that would otherwise be nontrivial to construct, as observed in [22].) In this case, if \(f:Q\odot M\rightarrow M\odot P\) and \(g:P\odot N \rightarrow N\odot L\) are two 2-cells, we have the composite
$$\begin{aligned} (\mathrm id _M\odot g)(f\odot \mathrm id _N):Q\odot M\odot N \overset{}{\longrightarrow }M\odot N\odot L. \end{aligned}$$
 

Proposition 7.5

In the above situation, we have
$$\begin{aligned} \mathrm{tr }\big ((\mathrm id _M\odot g)(f\odot \mathrm id _N)\big ) = \mathrm{tr }(g)\circ \mathrm{tr }(f). \end{aligned}$$
 
This equality is shown graphically in Fig. 20; again the deformation should be fairly easy to visualize.
Fig. 20

String diagram picture of Proposition 7.5 (color figure online)

In [32], this proposition was used to extend classical multiplicativity results for the Lefschetz number and Nielsen number to the Reidemeister trace.

Finally, recall that any 2-cell \(f:Q\otimes M \rightarrow M\otimes P\) has a mate \({{f}^{\bigstar }}:{{M}^{\bigstar }}\otimes Q \rightarrow P\otimes {{M}^{\bigstar }}\). Thus, in addition to calculating the trace of \(f\), we can use the analogous notion of trace for the left dualizable object \({{M}^{\bigstar }}\) to calculate the trace of \({{f}^{\bigstar }}\).

Proposition 7.6

If \(M\) is right dualizable and \(f:Q\odot M\rightarrow M\odot P\) is any 2-cell, then \(\mathrm{tr }(f)=\mathrm{tr }({{f}^{\bigstar }})\).

This equality is pictured graphically in Fig. 21.
Fig. 21

The trace of a mate (color figure online)

In [17], a list of similar properties of the canonical symmetric monoidal trace was used to define the notion of traced symmetric monoidal category. We could use the above properties to define an abstract notion of “traced bicategory”, but we have no use for such a definition at present.

8 Functoriality of trace in bicategories with shadows

We now move on to the crucial property of bicategorical trace for topological applications: its functoriality. As mentioned in the introduction, this is what implies refinements of the Lefschetz fixed point theorem; see Example 8.5.

First of all, we recall that a lax functor \(F:\fancyscript{B}\rightarrow \fancyscript{C}\) between bicategories consists of a function \(F_0\) from the objects of \(\fancyscript{B}\) to the objects of \(\fancyscript{C}\), functors
$$\begin{aligned} F_{R,S}:\fancyscript{B}(R,S)\rightarrow \fancyscript{C}(F_0(R),F_0(S)) \end{aligned}$$
and natural transformations
$$\begin{aligned} \mathfrak c :F_{R,S}(M)\odot F_{S,T}(N)&\overset{}{\longrightarrow }&F_{R,T}(M\odot N)\\ \mathfrak i :U_{F(R)}&\overset{}{\longrightarrow }&F(U_R) \end{aligned}$$
satisfying appropriate coherence axioms. If all maps \(\mathfrak i \) are isomorphisms, we call \(F\) normal, and if all maps \(\mathfrak c \) and \(\mathfrak i \) are isomorphisms, we call it a strong functor. We often abuse notation by writing simply \(F\) instead of \(F_0\) and \(F_{R,S}\).
We draw string diagram pictures for functors between bicategories similar to the way we draw them for monoidal functors, by superimposing a pattern on the strings and vertices of the diagram to which the functor is applied. However, we do not superimpose this pattern on a colored 0-cell region unless we wish to indicate application of the functor to a 1- or 2-cell bounded by the 0-cell in question. That is, if the 0-cell \(R\) in \(\fancyscript{B}\) is denoted by the color blue, then in the context of a functor \(F:\fancyscript{B}\rightarrow \fancyscript{C}\) where we are drawing string diagrams in \(\fancyscript{C}\), a blue region
(color figure online) will denote the 0-cell \(F(R)\)—there being nothing else it could mean, since \(R\) itself is not a 0-cell in \(\fancyscript{C}\). This potentially confusing convention is absolutely necessary, because otherwise we could not distinguish between the 1-cells \(F(M\odot N)\) and \(F M \odot F N\). With our convention, however, we can draw them as
   and   
respectively. The data and axioms of a lax functor can then be drawn as in Figs. 22 and 23. Note that as before, the final axiom is just naturality of \(\mathfrak c \).
Fig. 22

The data for a lax functor (color figure online)

 
Fig. 23

The axioms for a lax functor (color figure online)

 

Definition 8.1

[29] Let \(\fancyscript{B}\) and \(\fancyscript{C}\) be bicategories with shadow functors, whose target categories are \(\mathbf T \) and \(\mathbf Z \), respectively. A lax shadow functor is a lax functor \(F:\fancyscript{B}\rightarrow \fancyscript{C}\) together with a functor \(F_\mathrm{tr }:\mathbf T \rightarrow \mathbf Z \) and a natural transformation
$$\begin{aligned} \phi :\langle \langle - \rangle \rangle _{\fancyscript{C}} \circ F \overset{}{\longrightarrow }F_\mathrm{tr }\circ \langle \langle - \rangle \rangle _\fancyscript{B}\end{aligned}$$
(8.1)
such that the following diagram commutes whenever it makes sense.
If \(F\) is a strong functor and \(\phi \) is an isomorphism, we call \(F\) a strong shadow functor.
If \(F\) is a lax shadow functor, we depict the functor \(F_\mathrm{tr }\) by covering an entire cylinder in the pattern that denotes \(F\), and we depict the transformation \(\phi \) and its axioms as shown in Fig. 24.
Fig. 24

A lax shadow functor (color figure online)

 

Proposition 8.2

Let \(F:\fancyscript{B}\rightarrow \fancyscript{C}\) be a lax shadow functor and \(M:R\shortmid \!\!\!\rightarrow S\) a right dualizable 1-cell in \(\fancyscript{B}\).
  1. (i)

    If \(\mathfrak c :F(M)\odot F({{M}^{\bigstar }})\rightarrow F(M\odot {{M}^{\bigstar }})\) and \(\mathfrak i :U_F(R)\rightarrow F(U_R)\) are isomorphisms, then \(F(M)\) is dualizable with dual \(F({{M}^{\bigstar }})\).

     
  2. (ii)
    If, furthermore, \(\mathfrak c _{M,P}:F(M)\odot F(P)\rightarrow F(M\odot P)\) is an isomorphism, then for any \(f:Q\odot M\rightarrow M\odot P\), the following square commutes:
     

 

Proof

Statement (i) is proven exactly as in the symmetric monoidal case, while a graphical proof of statement (ii) is shown in Fig. 25. Since this is one of the centrally important facts about bicategorical traces, and since string diagrams for functors have not yet been formalized, we have chosen to show two intermediate steps of this proof. To get from the first diagram to the second, we apply naturality to slide an instance of \(\mathfrak c \) and of its inverse up to the top and bottom, respectively, and then apply the axioms from Fig. 23 to cancel them with an instance of \(\mathfrak i \) and its inverse, respectively. The step from the second to the third diagram is precisely the axiom in Fig. 24c, applied to the 1-cells \(P\odot {{M}^{\bigstar }}\) and \(M\). Finally, to get to the final diagram we cancel an instance of \(\mathfrak c \) with its inverse (removing the “hole” in the middle), and apply naturality to slide \(\phi \) all the way up to the top. \(\square \)

 
Fig. 25

Functoriality of the bicategorical trace (color figure online)

As in the symmetric monoidal case, in the situation of (i) above, we say that \(F\) preserves the dual \({{M}^{\bigstar }}\) of \(M\).

Example 8.3

If \(\mathbf C \) and \(\mathbf D \) are symmetric monoidal categories, and we equip \(\mathbf{C }/\!_\star \) and \(\mathbf{D }/\!_\star \) with their canonical shadows, then any lax monoidal functor \(F:\mathbf C \rightarrow \mathbf D \) gives a lax shadow functor \(\mathbf{C }/\!_\star \rightarrow \mathbf{D }/\!_\star \) such that \(\phi \) is an isomorphism. In this case Proposition 8.2 reduces to Proposition 2.7.

 

Example 8.4

Homology is a lax shadow functor
$$\begin{aligned} \mathcal C h /\!_\mathcal{R ing }\rightarrow \mathcal G r \mathcal M od /\!_\mathcal{R ing }, \end{aligned}$$
where \(\mathcal G r \mathcal M od /\!_\mathcal{R ing }\) is like \(\mathcal M od /\!_\mathcal{R ing }\) except that its 1-cells are graded bimodules. The Künneth theorem holds for rings that are not necessarily commutative, so Proposition 8.2 implies that \(\mathrm{tr }(H_*(f))=H_*(\mathrm{tr }(f))\) for any map \(f:Q_*\odot C_*\rightarrow C_*\odot P_*\) of a chain complex of \(S\)-\(R\)-modules, as long as \(C_p\) and \(H_p(C_*)\) are projective for each \(p\) and \(C_*\) is finitely generated.

 

Example 8.5

The rational chain complex functor induces a lax shadow functor
$$\begin{aligned} C_*:\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\rightarrow \mathrm Ho (\mathcal C h /\!_\mathcal{R ing }). \end{aligned}$$
On 0-cells we have \(C_*(G)=\mathbb Q G\), the group ring, while on 1-cells and 2-cells it is the usual rational chain complex functor, equipped with the induced actions by group rings. The shadow action \((C_*)_{\mathrm{tr }} :\mathrm Ho (\mathbf Sp )\rightarrow \mathrm Ho (\mathbf Ch _\mathbb{Z })\) is simply the ordinary rational chain complex.
If \(f:X\rightarrow X\) is an endomorphism of a closed smooth manifold or a compact ENR, as in Example 6.5, then
$$\begin{aligned} C_*(\tilde{f}):C_*(\tilde{X})\rightarrow C_*(\tilde{X}) \end{aligned}$$
is \(\psi \)-equivariant for the induced map \(\psi :\mathbb Q \pi _1(X)\rightarrow \mathbb Q \pi _1(X)\). Therefore, as in Example 5.3, we can regard it as a 2-cell (also denoted \(C_*(\tilde{f})\))
$$\begin{aligned} C_*(\tilde{X}) \rightarrow C_*(\tilde{X}) \odot \mathbb{Q \pi _1(X)}_{\psi }. \end{aligned}$$
Its trace in \(\mathrm Ho (\mathcal C h /\!_\mathcal{R ing })\) is then a map \(\mathbb Q \rightarrow \mathbb Q \langle \langle \pi _1(X)_\psi \rangle \rangle \). This is another way to define the Reidemeister trace; Proposition 8.2 shows that
$$\begin{aligned} \mathrm{tr }(C_*(\tilde{f})) = C_*(\mathrm{tr }(\tilde{f})). \end{aligned}$$
(Recall that \(\mathrm{tr }(\tilde{f})\) was what we called the Reidemeister trace in Example 6.5.) This is a more refined version of the Lefschetz fixed point theorem; see [29].

We might like to be able to combine this example with the previous one and calculate the Reidemeister trace at the level of homology, but unfortunately the resulting modules over the group ring are rarely projective, so the Künneth theorem generally fails.

 

9 2-functoriality of trace in bicategories with shadows

We observed in Sect. 2 that in the symmetric monoidal case, traces are respected not only by monoidal functors, but by monoidal transformations (Proposition 2.10). To conclude the main portion of the paper, we would like to prove a version of this for bicategories, but we have to be careful regarding what sort of transformation to consider. It turns out that the appropriate type is the following.

Definition 9.1

Let \(\fancyscript{B}\) and \(\fancyscript{C}\) be bicategories and \(F,G:\fancyscript{B}\rightarrow \fancyscript{C}\) be lax functors. A conjunctional transformation \(\alpha :F\rightarrow G\) consists of the following.
  1. (i)

    For each 0-cell \(R\in \fancyscript{B}\), a right dualizable 1-cell \(\alpha _{R}:F R \shortmid \!\!\!\rightarrow G R\) in \(\fancyscript{C}\), with right dual \(\alpha ^{\bigstar }_{R}\).

     
  2. (ii)

    For each 1-cell \(M:R\shortmid \!\!\!\rightarrow S\) in \(\fancyscript{B}\), a 2-cell \(\alpha _M:F(M) \odot \alpha _S \rightarrow \alpha _R \odot G(M)\) in \(\fancyscript{C}\), with consequent mate \(\alpha ^{\bigstar }_{M}:\alpha ^{\bigstar }_{R}\odot F(M) \rightarrow G(M)\odot \alpha ^{\bigstar }_{S}\).

     
  3. (iii)

    Some coherence axioms are satisfied. (See below for the axioms.)

     
 

 

Remark 9.2

To motivate this definition, we consider the main example we want to apply it to. Recall from Example 8.5 that we have a lax shadow functor
$$\begin{aligned} C_*(-;\mathbb Q ):\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\rightarrow \mathrm Ho (\mathcal C h /\!_\mathcal{R ing }) \end{aligned}$$
which enables us to compute the Reidemeister trace as a bicategorical trace in \(\mathrm Ho (\mathcal C h /\!_\mathcal{R ing })\). In Example 2.11 we used Proposition 2.10 to conclude that the ordinary Lefschetz number is the same whether computed with integral or rational coefficients, so we would like a similar statement in the bicategorical situation.
We certainly have another lax shadow functor
$$\begin{aligned} C_*(-;\mathbb Z ):\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\rightarrow \mathrm Ho (\mathcal C h /\!_\mathcal{R ing }) \end{aligned}$$
defined using integral chain complexes. (Since we are working only at the level of chain complexes, as in the remark at the end of Example 2.11, we don’t need to quotient by torsion to make this a strong functor.) Thus, when looking for a definition of transformation, we should ask what sort of transformation the inclusion \(\mathbb Z \rightarrow \mathbb Q \) induces from \(C_*(-;\mathbb Z )\) to \(C_*(-;\mathbb Q )\).

The first obvious thing that is induced is a collection of ring homomorphisms \(\alpha _G:\mathbb Z G \rightarrow \mathbb Q G\), for any 0-cell \(G\) in \(\mathcal S p /\!_\mathcal{G rp }\). A ring homomorphism \(\psi :R\rightarrow S\) is not itself any sort of cell in \(\mathcal C h /\!_\mathcal{R ing }\), but it does induce a dual pair of bimodules \((_\psi S,S_{\psi })\). (We have already met these bimodules in Example 5.3, and some analogous objects in Example 6.5, where they supplied the 1-cells \(P\) and \(Q\) by which bicategorical traces were “twisted.”) This motivates the choice to take dual pairs as the 1-cell components of a conjunctional transformation.

Next, for any 1-cell \(M:G\shortmid \!\!\!\rightarrow H\) in \(\mathcal S p /\!_\mathcal{G rp }\), we have a map
$$\begin{aligned} \alpha _M:C_*(M;\mathbb Z )\rightarrow C_*(M;\mathbb Q ) \end{aligned}$$
of chain complexes. This is not a 2-cell in any hom-category of \(\mathcal C h /\!_\mathcal{R ing }\), but instead is an “\(\alpha _G\)-\(\alpha _H\)-equivariant map” (i.e. it satisfies \(\alpha _M(x\cdot m \cdot y) = \alpha _G(g)\cdot \alpha _M(m) \cdot \alpha _H(y)\)). Such an equivariant map can be identified with a map
$$\begin{aligned} C_*(M;\mathbb Z ) \;\overset{}{\longrightarrow }\;_{i_G}\mathbb{Q G} \;\odot \; C_*(M;\mathbb Q ) \;\odot \; \mathbb{Q H}_{i_H}. \end{aligned}$$
Finally, the mate of such a map is a morphism
$$\begin{aligned} C_*(M;\mathbb Z ) \;\odot \; _{i_H}\mathbb{Q H} \;\overset{}{\longrightarrow }\; _{i_G}\mathbb{Q G} \;\odot \; C_*(M;\mathbb Q ) \end{aligned}$$
of \(\mathbb Z G\)-\(\mathbb Q H\)-bimodules, and this provides the 2-cell components of a conjunctional transformation.

As we will see later, this seemingly ad hoc definition also provides exactly the right structure necessary to prove 2-functoriality of traces. It can also be shown to arise naturally from a natural sort of transformation for a class of “fibrant” double categories; see [35, 36].

As usual, the coherence axioms of a conjunctional transformation are most naturally visualized in string diagram notation. Just like in the symmetric monoidal case, we picture such a transformation as a ‘membrane’ dividing the \(F\)-region from the \(G\)-region. However, now the membrane itself is actually a string: a horizontally drawn string representing the dual pair \((\alpha _{R},\alpha ^{\bigstar }_{R})\), as described in Sect. 5. Similarly, the locations where other strings cross over the membrane represent the 2-cell components of \(\alpha \), although we usually do not draw them as nodes. Note that according to the convention for drawing functors established in Sect. 8, a colored region with no pattern can equally denote a 0-cell in the image of \(F\) or in the image of \(G\), although in practice there should never be any ambiguity about which is intended.

With these conventions in place, Fig. 26 displays the data of a conjunctional transformation, and Fig. 27 shows the coherence axioms.
Fig. 26

The data of a conjunctional transformation (color figure online)

Fig. 27

The axioms of a conjunctional transformation (color figure online)

 

When interpreting the final equation in Fig. 27, note that according to the conventions for functors established in Sect. 8,
denotes \(\alpha _{M\odot N}\), while
denotes \(\left(\alpha _M \odot \mathrm id _{G(N)}\right)\circ \left(\mathrm id _{F(M)}\odot \alpha _N\right)\).

Remark 9.3

The axioms of a conjunctional transformation say precisely that the 1- and 2-cell components \(\alpha _R\) and \(\alpha _M\) form an “oplax natural transformation,” or equivalently that the dual components \(\alpha ^{\bigstar }_{R}\) and \(\alpha ^{\bigstar }_{M}\) form a “lax natural transformation.” We call this a conjunctional transformation because it is a “conjoint pair” in the double category of lax and oplax natural transformations; see for instance [35].

Before we can state and prove an analogue of Proposition 2.10, we need one further observation. Namely, if \(Q:R\shortmid \!\!\!\rightarrow R\) is an endo-1-cell, then since \(\alpha _R\) is right dualizable, we can take the trace of the 2-cell \(\alpha _Q:F(Q) \odot \alpha _R \rightarrow \alpha _R \odot G(Q)\) to obtain a morphism \(\mathrm{tr }(\alpha _Q):\langle \langle F(Q) \rangle \rangle \rightarrow \langle \langle G(Q) \rangle \rangle \). We naturally depict this as in Fig. 28.
Fig. 28

The trace of a conjunctional transformation (color figure online)

Definition 9.4

Let \(\fancyscript{B}\) and \(\fancyscript{C}\) be bicategories with shadows, \(F, G:\fancyscript{B}\rightarrow \fancyscript{C}\) lax functors, and \(\alpha :F\rightarrow G\) a conjunctional transformation, let \(M:R\shortmid \!\!\!\rightarrow S\) be right dualizable in \(\fancyscript{B}\), and suppose that \(F\) and \(G\) preserve its dual \({{M}^{\bigstar }}\).
  1. (i)

    Then \(\alpha _M:F(M) \odot \alpha _S \rightarrow \alpha _R \odot G(M)\) is an isomorphism, whose inverse is the mate of \(\alpha _{{{M}^{\bigstar }}}\) under the dual pairs \((F(M),F({{M}^{\bigstar }}))\) and \((G(M),G({{M}^{\bigstar }}))\). Assume furthermore that \(\mathfrak c :F(M)\odot F(P)\rightarrow F(M\odot P)\) and \(\mathfrak c :G(M)\odot G(P)\rightarrow G(M\odot P)\) are isomorphisms.

     
  2. (ii)
    Then for any 2-cell \(f:Q\odot M\rightarrow M\odot P\), the following square commutes.
     

 

Proof

For (i), Fig. 29 displays the equality asserting that one composite of \(\alpha _M\) with its putative inverse is the identity. To prove it we simply “pull the lower loop through the \(\alpha \) membrane,” using the axioms from Fig. 27 (and the consequent relations for the inverses of \(\mathfrak c \) and \(\mathfrak i \), when these transformations are invertible) and then straighten the \(F(M)\) string using a triangle identity. The other equality is similar. (This part does not require \(\alpha \) to be conjunctional, only oplax. A corresponding result is true when \(\alpha \) is lax and \(M\) is left dualizable.)
Fig. 29

Duals invert a bicategorical transformation (color figure online)

The proof of (ii) is the first (and only) time in the paper that we do not need to invoke the actual definition of the bicategorical trace: all we need are the properties proven in Sect. 7 and the axioms of a conjunctional transformation. Thus, we can now simplify the pictures by drawing the bicategorical trace with a plain loop around the back of the cylinder, rather than breaking it down into coevaluation, shadow, and evaluation. With this convention, the proof is shown in Fig. 30. In the first step, we apply Proposition 7.5 to replace \(\mathrm{tr }(\alpha _P)\circ \mathrm{tr }(\mathfrak c ^{-1}\circ Ff\circ \mathfrak c )\) by the trace of a composite. Secondly, we use part (i) to introduce an instance of \(\alpha _M\) and its inverse. In the third step, we use the axioms of Fig. 27 to slide the \(\alpha \) string across \(\mathfrak c ^{-1}\circ Ff\circ \mathfrak c \). We then use Proposition 7.2 to bring \(\alpha _M\) around the back of the cylinder, so that we can cancel it with its inverse. Finally, we apply Proposition 7.5 again to obtain \(\mathrm{tr }(\mathfrak c ^{-1}\circ Gf\circ \mathfrak c )\circ \mathrm{tr }(\alpha _Q)\), as desired.

\(\square \)

Fig. 30

2-functoriality of the bicategorical trace (color figure online)

When \(F\) and \(G\) are additionally shadow functors, we can also compare \(F_\mathrm{tr }(\mathrm{tr }(f))\) and \(G_\mathrm{tr }(\mathrm{tr }(f))\), but we need some extra structure on \(\alpha .\)

Definition 9.5

Let \(F,G:\fancyscript{B}\rightarrow \fancyscript{C}\) be lax shadow functors. A shadow conjunctional transformation \(\alpha :F\rightarrow G\) is a conjunctional transformation together with a natural transformation
$$\begin{aligned} \alpha _{tr} :F_{tr} \rightarrow G_{tr} \end{aligned}$$
such that a coherence axiom is satisfied.
We draw the component
$$\begin{aligned} (\alpha _{tr})_{\langle \langle M \rangle \rangle }:F_{tr}\big (\langle \langle M \rangle \rangle \big ) \rightarrow G_{tr}\big (\langle \langle M \rangle \rangle \big ) \end{aligned}$$
as in Fig. 31. Of course, despite what the picture may suggest, \(\alpha _{tr}\) is not actually the trace of anything.
Fig. 31

A shadow conjunctional transformation (color figure online)

We can now observe that if \(\alpha _{tr}\) is a shadow conjunctional transformation, \(M\) is right dualizable, \(f:Q\odot M \rightarrow M \odot P\) is a 2-cell, and Proposition 8.2 applies to \(F\) and to \(G\), then all six faces of the cube shown in Fig. 32 commute. The back face is 9.4, the front is just naturality of \(\alpha _{tr}\), the top and bottom faces are 8.2 for \(F\) and \(G\) respectively, and the left and right faces are the axiom of \(\alpha _{tr}\) from Fig. 31b.
Fig. 32

The cube of traces

 

Example 9.6

As observed in Remark 9.2, we have two lax shadow functors \(C_*(-;\mathbb Z )\) and \(C_*(-;\mathbb Q )\) from \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\) to \(\mathrm Ho (\mathcal C h /\!_\mathcal{R ing })\), and the inclusion \(\mathbb Z \rightarrow \mathbb Q \) then defines a conjunctional transformation from the first to the second. For any \(P:R\shortmid \!\!\!\rightarrow R\) in \(\mathrm Ho (\mathcal S p /\!_\mathcal{G rp })\), the shadow \(\mathrm{tr }(\alpha _P)\) is just the induced map \(\langle \langle C_*(P;\mathbb Z ) \rangle \rangle \rightarrow \langle \langle C_*(P;\mathbb Q ) \rangle \rangle \). Therefore, Theorem 9.4 implies that this map identifies the Reidemeister trace, computed as in Example 8.5 using the rational chain complex, with the analogous version computed using the integral chain complex.

 

This transformation \(\alpha \) is also a shadow conjunctional transformation; the map \(\alpha _{tr}:C_*(-;\mathbb Z )_{\mathrm{tr }} \rightarrow C_*(-;\mathbb Q )_{\mathrm{tr }}\) simply includes the ordinary integral chain complex of a spectrum into its rational one. Thus we have four different ways to compute the Reidemeister trace, and the cube in Fig. 32 shows that they are all equivalent.

 

Notes

Acknowledgments

The authors would like to thank Niles Johnson, for careful reading and helpful comments.

References

  1. 1.
    Adams, J.F.: Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture. In: Algebraic topology, Aarhus (1982). (Aarhus 1982), Lecture Notes in Mathematics, vol. 1051, pp. 483–532. Springer, Berlin (1984)Google Scholar
  2. 2.
    Baez, J.C., Dolan, J.: Higher-dimensional algebra. III. \(n\)-categories and the algebra of opetopes. Adv. Math. 135(2), 145–206 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bénabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar, pp. 1–77. Springer, Berlin (1967)Google Scholar
  4. 4.
    Brown, R.F.: The Lefschetz fixed point theorem. Scott, Foresman and Co., Glenview (1971)Google Scholar
  5. 5.
    Day, B., Street, R.: Monoidal bicategories and Hopf algebroids. Adv. Math. 129(1), 99–157 (1997)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    tom Dieck, T.: Transformation groups, de Gruyter Studies in Mathematics, vol. 8. Walter de Gruyter & Co., Berlin (1987)Google Scholar
  7. 7.
    Dold, A.: The fixed point index of fibre-preserving maps. Invent. Math. 25, 281–297 (1974)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dold, A., Puppe, D.: Duality, trace, and transfer. In: Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), pp. 81–102. PWN, Warsaw (1980)Google Scholar
  9. 9.
    Freyd, P.J., Yetter, D.N.: Braided compact closed categories with applications to low-dimensional topology. Adv. Math. 77(2), 156–182 (1989)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hasegawa, M.: The uniformity principle on traced monoidal categories. Publ. Res. Inst. Math. Sci. 40(3), 991–1014 (2004). http://projecteuclid.org/getRecord?id=euclid.prims/1145475500
  11. 11.
    Hattori, A.: Rank element of a projective module. Nagoya Math. J. 25, 113–120 (1965)MathSciNetMATHGoogle Scholar
  12. 12.
    Jiang, B.J.: Lectures on Nielsen Fixed Point Theory, Contemporary Mathematics, vol. 14. American Mathematical Society, Providence (1983)CrossRefGoogle Scholar
  13. 13.
    Joyal, A., Street, R.: The geometry of tensor calculus. II. Draft available at http://www.math.mq.edu.au/street/GTCII.pdf
  14. 14.
    Joyal, A., Street, R.: Planar diagrams and tensor algebra. Draft available at http://www.math.mq.edu.au/street/PlanarDiags.pdf
  15. 15.
    Joyal, A., Street, R.: The geometry of tensor calculus. I. Adv. Math. 88(1), 55–112 (1991)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119(3), 447–468 (1996)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. J. Pure Appl. Algebra 19, 193–213 (1980)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kelly, G.M., Street, R.: Review of the elements of \(2\)-categories. In: Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Mathematics, vol. 420, pp. 75–103. Springer, Berlin (1974)Google Scholar
  20. 20.
    Lewis Jr., L.G., May, J.P., Steinberger, M., McClure, J.E.: Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213. Springer-Verlag, Berlin (1986). With contributions by J. E. McClureGoogle Scholar
  21. 21.
    May, J.P.: The additivity of traces in triangulated categories. Adv. Math. 163(1), 34–73 (2001)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    May, J.P., Sigurdsson, J.: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132. American Mathematical Society, Providence (2006)Google Scholar
  23. 23.
    McCurdy, M.: Strings and stripes: graphical calculus for monoidal functors and monads http://web.science.mq.edu.au/mmccurdy/cms2010talk.pdf (2010). Talk at CMS meeting in Fredricton, NB
  24. 24.
    McCurdy, M.B.: Graphical methods for tannaka duality of weak bialgebras and weak hopf algebras in arbitrary braided monoidal categories. arXiv:1110.5542Google Scholar
  25. 25.
    Penrose, R.: Applications of negative dimensional tensors. In: Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 221–244. Academic Press, London (1971)Google Scholar
  26. 26.
    Penrose, R., Rindler, W.: Spinors and space-time. vol. 1. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984). Two-spinor calculus and relativistic fieldsGoogle Scholar
  27. 27.
    Ponto, K.: Coincidence invariants and higher Reidemeister traces. arXiv:1209.3710Google Scholar
  28. 28.
    Ponto, K.: Equivariant fixed point theory. arXiv:0910.1274Google Scholar
  29. 29.
    Ponto, K.: Fixed point theory and trace for bicategories. Astérisque (333), xii+102 (2010). arXiv:0807.1471Google Scholar
  30. 30.
    Ponto, K.: Relative fixed point theory. Algebr. Geom. Topol. 11(2), 839–886 (2011). doi: 11110.2140/agt.2011.11.839. http://dx.doi.org/10.2140/agt.2011.11.839
  31. 31.
    Ponto, K., Shulman, M.: Duality and trace in indexed monoidal categories. Submitted. Available online at http://www.math.ucsd.edu/mshulman/papers/indexed-color.pdf
  32. 32.
    Ponto, K., Shulman, M.: The multiplicativity of fixed point invariants. arXiv:1203.0950Google Scholar
  33. 33.
    Ranicki, A.: The algebraic theory of surgery. II. Applications to topology. Proc. London Math. Soc. (3) 40(2), 193–283 (1980)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Selinger, P.: A survey of graphical languages for monoidal categories. In: B. Coeke (ed.) New Structures for Physics, chap. 4. Springer, Berlin (2011). Available at http://www.mscs.dal.ca/selinger/papers.html#graphical. arXiv:0908.3347
  35. 35.
    Shulman, M.: Constructing symmetric monoidal bicategories. arXiv:1004.0993Google Scholar
  36. 36.
    Shulman, M.: Framed bicategories and monoidal fibrations. Theory Appl. Categ. 20(18), 650–738 (electronic) (2008). arXiv:0706.1286Google Scholar
  37. 37.
    Stallings, J.: Centerless groups—an algebraic formulation of Gottlieb’s theorem. Topology 4, 129–134 (1965)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Street, R.: Limits indexed by category-valued \(2\)-functors. J. Pure Appl. Algebra 8(2), 149–181 (1976)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Street, R.: Low-dimensional topology and higher-order categories I. In: Proceedings of CT95, Halifax (1995). Available at http://www.mta.ca/cat-dist/CT95Docs/LowDim.ps
  40. 40.
    Street, R.: Categorical structures. In: Handbook of algebra, vol. 1, pp. 529–577. North-Holland, Amsterdam (1996). doi: 10.1016/S1570-7954(96),80019-2. http://dx.doi.org/10.1016/S1570-7954(96)80019-2

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

Personalised recommendations