Shadows and traces in bicategories
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Abstract
Traces in symmetric monoidal categories are wellknown 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 HattoriStallings 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 2functoriality, which are essential to its applications in fixedpoint theory. Throughout we make use of an appropriate “cylindrical” type of string diagram, which we justify formally in an appendix.
Keywords
Bicategory Trace Fixedpoint theoryMathematics Subject Classification (2000)
18D05 55M201 Introduction
The purpose of this paper is to study and exposit a categorical notion of trace for endo2cells 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 wellknown 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 categorytheoretic 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 finitedimensional 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 categorytheoretic expressions of other fixedpoint 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 fixedpoint 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 fixedpoint–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 “HattoriStallings 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 HattoriStallings trace. It was invented by the first author for applications in fixed point theory; see [29].
 (i)
Rings and bimodules (the HattoriStallings context), as well as generalizations such as DGAs and chain complexes.
 (ii)
Parametrized spaces and parametrized spectra, as studied in [22].
 (iii)
Spaces with group actions, as studied in [33].
 (iv)
\(n\)dimensional manifolds and cobordisms.
 (v)
Categories and profunctors (also called bimodules or distributors), with enriched and internal variations.
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 categorytheoretic ideas. On the other hand, there may be categorytheorists 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 categorytheoretic, 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 nontopological 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 allimportant functoriality. Less often cited, but also important, is what we call “2functoriality”: 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 2functoriality 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 1cells. 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 deformationinvariant composite. This is a technical, but fairly straightforward extension of the classical proof for monoidal categories in [15].
2 Traces in symmetric 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
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.
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
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 \(Q=I\)).
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 finitedimensional, 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 quasiisomorphic 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 SpanierWhitehead 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
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.
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
 (i)
Then \(F(M)\) is dualizable with dual \(F({{M}^{\bigstar }})\).
 (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)
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 \(\mathrm{tr }(f\otimes _RS)=\mathrm{tr }(f)\otimes _RS\).
Proposition 2.10
 (i)
\(\alpha _M:F(M)\rightarrow G(M)\) is an isomorphism, whose inverse is the mate of \(\alpha _{{{M}^{\bigstar }}}\), and
 (ii)For any \(f:Q\otimes M \rightarrow M\otimes P\), the following square commutes.
 (iii)
In particular, for \(f:M\rightarrow M\), we have \(\mathrm{tr }(F(f)) = \mathrm{tr }(G(f))\).
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 noncommutative situations, where we don’t even have a monoidal category, let alone a symmetric one. For example, if \(R\) is a noncommutative 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 collection of objects or 0cells \(R,S,T,\ldots \).

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 isomorphismssatisfying the same coherence axioms as for a monoidal category.$$\begin{aligned} \mathfrak a :M\odot (N\odot P)&\overset{\cong }{\longrightarrow } (M\odot N)\odot P\\ \mathfrak l :U_R \odot M&\overset{\cong }{\longrightarrow } M\\ \mathfrak r :M\odot U_S&\overset{\cong }{\longrightarrow } M \end{aligned}$$
We call the objects of \(\fancyscript{B}(R,S)\) 1cells and its morphisms 2cells. We regard 1cells and 2cells in a bicategory as analogous to the objects and morphisms in a monoidal category, respectively, with the 0cells playing a “bookkeeping” role informing us which 1cells can be tensored with which others. We often write a 1cell \(M\in \fancyscript{B}(R,S)\) as \(M:R \shortmid \!\!\!\rightarrow S\) to distinguish it from a 2cell \(f:M\rightarrow N\).
Example 3.2
There is a bicategory \(\mathcal M od /\!_\mathcal{R ing }\) whose objects are notnecessarilycommutative 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 1cells. 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 1cells are modules, while the name \(\mathbf{C }/\!_\star \) indicates that there is a unique object \(\star \) and the 1cells are the objects of \(\mathbf C \).
Remark 3.5
We have chosen to write the composite of 1cells \(M:R \shortmid \!\!\!\rightarrow S\) 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 1cells are functors, and whose 2cells 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 \(G \circ F:A\rightarrow C\).
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 1cells \(M:R \shortmid \!\!\!\rightarrow S\) 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 1cell \(R \shortmid \!\!\!\rightarrow R\) and the other is a 1cell \(S \shortmid \!\!\!\rightarrow S\). 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
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
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 will consider more examples in Sect. 6 after defining traces.
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 wellspecified 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.)
5 Duality and trace
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.
There is an evident dual construction for left dualizable objects, and the triangle identities show that the two are inverses. Therefore, giving a 2cell \(Q\odot M\rightarrow N\odot P\) is equivalent to giving a 2cell \({{N}^{\bigstar }}\odot Q\rightarrow P\odot {{M}^{\bigstar }}\). 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 2cell 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
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 1cells 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 HattoriStallings 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 2cell \(M\rightarrow M\odot R_\psi \) 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 1category, shadow is a cyclic functor on endo1cells 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 1cells \(\mathbb Z \rightarrow R\) 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 HattoriStallings trace of the \(R\)module homomorphism \((s\cdot ):M\rightarrow M\) (which descends to \(\langle \langle S \rangle \rangle \) because the HattoriStallings 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 HattoriStallings trace of \(m\mapsto f(sm)=s\cdot f(m)\).
In particular, let \(G\) be a group, \(k\) a field, and \(V\) a finitedimensional left \(k G\)module (that is, a representation of \(G\)). If we regard \(V\) as a 1cell \(k G\rightarrow k\) 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
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 1cell \(R\!\xleftarrow {{{\ {h}\ }}}\! M\!\xrightarrow {\ g\ }\! S\) is right dualizable if and only if \(h\) is an isomorphism. If \(R\xleftarrow {{{\ \text{ id}\ }}} R\xrightarrow {\ g\ } S\) is such a dualizable 1cell, 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 \((n1)\)dimensional manifolds, whose 1cells are \(n\)dimensional cobordisms, and whose 2cells are boundarypreserving 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 2cells 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 2cells considered as “weak equivalences.” The theory of homotopy bicategories is not welldeveloped, 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 1cells 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 quasiisomorphisms of chain complexes). The composition \(\odot \) is now the derived functor of the tensor product of bimodules, and the shadow of a 1cell \(M:A\rightarrow A\) 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
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 1cell \(1\rightarrow G\) 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.
This trace is known as the Reidemeister trace. It is a refinement of the fixedpoint index, which separates out contributions coming from different “fixedpoint 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 fixedpoint 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 1cell \(R\shortmid \!\!\!\rightarrow S\) 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 pointsetlevel 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 1cell \(\hat{M}:B \shortmid \!\!\!\rightarrow \star \) or \(\check{M}:\star \shortmid \!\!\!\rightarrow B\) in \(\mathrm Ho (\mathcal S p /\!_\mathcal{T op })\). The 1cell \(\hat{M}\) 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 CostenobleWaner duality, is very different. For example, if the fiberwise sphere spectrum \(S_B= \Sigma ^{\infty }_{B,+}(B)\) is CostenobleWaner 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 CostenobleWaner duality also knows about the base space.
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 fixedpoint 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 CostenobleWaner 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
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 wellknown 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
Proposition 7.2
Corollary 7.3
If \(M\) and \(N\) are right dualizable and \(f:M\rightarrow N\) and \(g:N\rightarrow M\) are 2cells, 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 2cell, 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 1cells are represented by empty space.
Proposition 7.5
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 2cell \(f:Q\otimes M \rightarrow M\otimes P\) 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 2cell, then \(\mathrm{tr }(f)=\mathrm{tr }({{f}^{\bigstar }})\).
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.
Definition 8.1
Proposition 8.2
 (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 \(F({{M}^{\bigstar }})\).
 (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 1cells \(P\odot {{M}^{\bigstar }}\) 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. \(\square \)
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
Example 8.5
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 2functoriality 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
 (i)
For each 0cell \(R\in \fancyscript{B}\), a right dualizable 1cell \(\alpha _{R}:F R \shortmid \!\!\!\rightarrow G R\) in \(\fancyscript{C}\), with right dual \(\alpha ^{\bigstar }_{R}\).
 (ii)
For each 1cell \(M:R\shortmid \!\!\!\rightarrow S\) in \(\fancyscript{B}\), a 2cell \(\alpha _M:F(M) \odot \alpha _S \rightarrow \alpha _R \odot G(M)\) in \(\fancyscript{C}\), with consequent mate \(\alpha ^{\bigstar }_{M}:\alpha ^{\bigstar }_{R}\odot F(M) \rightarrow G(M)\odot \alpha ^{\bigstar }_{S}\).
 (iii)
Some coherence axioms are satisfied. (See below for the axioms.)
Remark 9.2
The first obvious thing that is induced is a collection of ring homomorphisms \(\alpha _G:\mathbb Z G \rightarrow \mathbb Q G\), for any 0cell \(G\) 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 1cells \(P\) and \(Q\) by which bicategorical traces were “twisted.”) This motivates the choice to take dual pairs as the 1cell components of a conjunctional transformation.
As we will see later, this seemingly ad hoc definition also provides exactly the right structure necessary to prove 2functoriality 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 2cell 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 0cell in the image of \(F\) or in the image of \(G\), although in practice there should never be any ambiguity about which is intended.
Remark 9.3
The axioms of a conjunctional transformation say precisely that the 1 and 2cell 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].
Definition 9.4
 (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.
 (ii)Then for any 2cell \(f:Q\odot M\rightarrow M\odot P\), the following square commutes.
Proof
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.
\(\square \)
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
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.
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