Abstract
Like categories, small 2categories have wellunderstood classifying spaces. In this paper, we deal with homotopy types represented by 2diagrams of 2categories. Our results extend lower categorical analogues that have been classically used in algebraic topology and algebraic Ktheory, such as the homotopy invariance theorem (by Bousfield and Kan), the homotopy colimit theorem (Thomason), Theorems A and B (Quillen), or the homotopy cofinality theorem (Hirschhorn).
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1 Introduction
If C is any small 2category, by replacing its homcategories C(x, y) by their classifying spaces C(x, y), one obtains a topological category whose Segal classifying space [30] is, by definition, the classifying space \(\!C \!\) of the 2category C. By this assignment \(C\mapsto \!C\!\), for example, 2groupoids (or, equivalently, crossed modules in the sense of Brown and Higgins) correspond to homotopy 2types (i.e., to CWcomplexes whose n \(\mathrm {th}\) homotopy groups at any base point vanish for \(n\ge 3\)) [8, 28], and, up to group completion, monoidal categories (regarded as 2categories with only one object) correspond to delooping spaces of the classifying spaces of their underlying categories [31]. A natural interpretation of the classifying space \(\!C\!\), in terms of charted Cbundles, is given in [3].
The category \(\mathbf {2}\mathbf {Cat}\) of (small) 2categories and 2functors has a Thomasonstyle model structure,^{Footnote 1} such that the classifying space functor \(C\mapsto \!C\!\) is an equivalence of homotopy theories between 2categories and topological spaces. This fact was first announced in [35, Theorem 4.5.1], but fully proved in [1, Théorème 6.27].^{Footnote 2} In [14, Theorem 4.5 (ii)], an extension of the celebrated Thomason’s homotopy colimit theorem [34, Theorem 1.2] asserts that, for any category C and functor \(\mathcal {D}:C\rightarrow \mathbf {2}\mathbf {Cat}\), the 2category \(\int _{C}\mathcal {D}\), obtained by applying the Grothendieck construction on \(\mathcal {D}\), is a model for its homotopy colimit (with respect to the Thomason model structure in \(\mathbf {2}\mathbf {Cat}\)). In other words, the induced functor between the homotopy categories \(\int _C :\mathrm {Ho}(\mathbf {2}\mathbf {Cat}^C)\rightarrow \mathrm {Ho}(\mathbf {2}\mathbf {Cat})\) is left adjoint to the functor assigning to any 2category the constant functor on C that it defines.
This paper focuses on the study of “homotopy colimits” of 2functors \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), from an indexing 2category C into the 2category \({\underline{\mathbf{2Cat}}}\) of 2categories, 2functors, and 2natural transformations, and our results extend lower categorical analogues classically used in algebraic topology and algebraic Ktheory. We shall stress that the main difference with the case of diagrams \(\mathcal {D}:C\rightarrow \mathbf {2}\mathbf {Cat}\), where C is a category, is that now there are 2cells \(\phi :p\Rightarrow p'\) in C that produce 2natural transformations \(\mathcal {D}\phi :\mathcal {D}p\Rightarrow \mathcal {D}p'\), and therefore homotopies between the induced maps on classifying spaces by the associated 2functors \(\mathcal {D}p\) and \(\mathcal {D}p'\), which must be taken into account. However, a warning is needed since, for now, there is no known model structure in the 2category \({\underline{\mathbf{2Cat}}}\) having the Thomason model structure in the underlying category \( \mathbf {2}\mathbf {Cat}\). So we are not actually correct to speak of (\(\mathbf {Cat}\)enriched) homotopy colimits of such 2functors \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\). However, as we shall prove in the paper, the Grothendieck construction 2functor
enjoys so many desirable homotopy properties, with respect to the Thomason model structure on \( \mathbf {2}\mathbf {Cat}\), that it deserves to be thought of as the correct “homotopy colimit” construction on 2diagrams of 2categories.
Interesting 2diagrams of 2categories naturally arise from basic problems in homotopy theory of 2categories. For example, the analysis of the homotopy fibres of the map \(\!F\!:\!A\!\rightarrow \!C\!\) induced on classifying spaces by a 2functor \(F:A\rightarrow C\) leads to the study of the 2functor \(F\!\downarrow \!: C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), which associates to each object \(c\in C\) the comma 2category \(F\!\downarrow \!c\), whose objects are 1cells in C of the form \(p:Fa\rightarrow c\). A relevant result here is the extension of Quillen’s Theorem B for 2functors in [14, Theorem 3.2], which assures us that, under reasonable necessary conditions, the sequences \(F\!\downarrow \!c\hookrightarrow A\rightarrow B\) are homotopyfibre sequences in \(\mathbf {2}\mathbf {Cat}\) (with respect to the Thomason model structure). See also the relative Quillen’s Theorem A for 2functors in [18, Théorèm 2.34].
There is another interesting source for 2diagrams of 2categories: The study and classification of opfibred 2categories. The wellknown Grothendieck correspondence between covariant pseudofunctors and opfibred categories [24] has been generalized to bicategories in [4, 9], where the authors prove, in particular, that the Grothendieck construction on 2functors \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) gives rise to 2categories \(\int _C\mathcal {D}\) endowed with a split 2opfibration over C, and this correspondence \(\mathcal {D}\mapsto \int _C\mathcal {D}\) is the function on objects of an equivalence between the 3category of 2functors \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) and the 3category of split 2opfibred 2categories over C.
The plan of the paper is as follows. After this introductory section, the paper is organized into five sections. Section 2 is preliminary and comprises some notations and a brief review of facts used later. This is itself subdivided into four sections with summaries concerning geometric realizations of bisimplicial sets, classifying spaces of 2categories, the Grothendieck construction on 2functors valued in \({\underline{\mathbf{2Cat}}}\), and the homotopyfibre 2functors respectively. Section 3 includes a main result of the paper, which we call the Homotopy Colimit Theorem for 2diagrams of 2categories. With Thomason’s homotopy colimit theorem as its natural precedent, the result therein states that, for any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the geometric realization of 2category \(\int _C\mathcal {D}\) has the same homotopy type as the simplicial 2category (i.e., simplicial object in \(\mathbf {2}\mathbf {Cat}\))
obtained by applying the bar (Borel or simplicial replacement) construction on \(\mathcal {D}\); that is, the simplicial 2category whose 2category of psimplices is
Hence, both constructions \(\mathcal {B}_C\) and \(\int _C\) can be interchanged for homotopy purposes. The proof of this theorem is quite long and technical, but the result is crucial and many further results are based on it. Section 3 also includes the Homotopy Invariance Theorem, stating that if \(\mathcal {D}\rightarrow \mathcal {E}\) is a 2transformation that is locally a weak equivalence, then the induced \(\int _C\mathcal {D}\rightarrow \int _C\mathcal {E}\) is also a weak equivalence. In Sect. 4, we deal with questions such as: When does a 2transformation \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\), between 2functors \(\mathcal {D},\mathcal {E}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), induce a homotopy left cofinal 2functor \(\int _C\Gamma :\int _C\mathcal {D}\rightarrow \int _C\mathcal {E}\)? Or when are the canonical squares (\(c\in \mathrm {Ob}C\), \(y\in \mathrm {Ob}\mathcal {E}_c\))
homotopy pullbacks? Our main results here are actually extensions of the wellknown Quillen’s Theorems A and B for functors between categories to morphisms between 2diagrams of 2categories. The final Sect. 5 is dedicated to analyzing the behavior of the Grothendieck construction when a 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) is composed with a 2functor \(F:A\rightarrow C\). There is a canonical 2functor \(\int _AF^*\mathcal {D}\rightarrow \int _C\mathcal {D}\), and we mainly study when this 2functor is a weak equivalence or, more interestingly, when the canonical pullback square in \(\mathbf {2}\mathbf {Cat}\)
is a homotopy pullback.
2 Preliminaries
This section aims to make this paper as selfcontained as possible; hence, at the same time as fixing notations and terminology, we review some needed constructions and facts concerning geometric realizations of bisimplicial spaces (Sect. 2.1), classifying spaces of (small) 2categories (Sect. 2.2), the Grothendieck construction on 2functors valuated in the 2category of 2categories (Sect. 2.3), and the homotopyfibre 2functors (Sect. 2.4). Although most of the material here is perfectly standard by now, so the expert reader may skip it, some notational conventions may be a bit idiosyncratic for some readers.
2.1 Geometric realizations of bisimplicial sets
For the general background on simplicial sets, refer to [7, 22]. As usually, \(\Delta \) denotes the category of finite ordered sets \([p]=\{0,\ldots ,p\}\) with nondecreasing maps between them as morphisms. Recall that it is generated by the injections \(d^i:[n1]\rightarrow [n]\) (cofaces), \(0\!\le i\!\le n\), which omit the ith element and the surjections \(s^i:[n+1]\rightarrow [n]\) (codegeneracies), \(0\!\le i\!\le n\), which repeat the ith element, subject to the wellknown cosimplicial identities: \(d^j d^i=d^i d^{j1}\) if \(i\!<j\), etc. Thus, in order to define a simplicial object in a category \(\mathcal {E}\), say \(S:\Delta ^{\!\mathrm {op}}\rightarrow \mathcal {E}\), it suffices to give the objects (of nsimplices) \(S_n\), \(n\ge 0\), together with morphisms
satisfying the wellknown basic simplicial identities: \(d_id_j=d_{j1}d_i\) if \(i\!<j\), etc.
In [30], Segal extended Milnor’s geometric realization process for simplicial sets to simplicial (compactly generated Hausdorff topological) spaces. The realization S of a simplicial space \(S:\Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {Top}\), is as follows: for each \(n\ge 0\), let \(\Delta ^n\) denote the standard ndimensional affine simplex and for each map \(\alpha :[n]\rightarrow [m]\) in \(\Delta \) let \(\alpha _*:\Delta ^n\rightarrow \Delta ^m\) be the induced affine map (i.e., \(\alpha _*(t_0,\ldots ,t_n)=(t_0^\prime ,\ldots ,t_m^\prime )\) with \(t_i^\prime =\sum _{\alpha (j)=i}t_j\)). Then the space S is defined from the topological sum \(\coprod _{n\ge 0} S_n\times \Delta ^n\) by identifying \((S\alpha (x),t)\in S_n\times \Delta ^n\) with \((x,\alpha _*t)\in S_m\times \Delta ^m\), for all \(x\in S_m\), \(t\in \Delta ^n\), and \(\alpha :[n]\rightarrow [m]\) in \(\Delta \). For instance, by regarding a set as a discrete space, the (Milnor’s) geometric realization of a simplicial set \(S:\Delta ^{\!{^\mathrm {op}}}\rightarrow \mathbf {Set}\) is S. A weak homotopy equivalence of simplicial sets is a simplicial map whose geometric realization is a homotopy equivalence or, equivalently, induces isomorphisms in homotopy groups.
A bisimplicial set is a functor \(S:\Delta ^{\!{\mathrm {op}}}\times \Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {Set}\). This amounts to a family of sets \(\{S_{p,q};\, p,q\ge 0\}\) together with horizontal and vertical face and degeneracy operators
with \(0\!\le i\!\le p\) and \(0\!\le j\!\le q\) respectively, such that, for all p and q, both \(S_{p,\bullet }\) and \(S_{\bullet ,q}\) are simplicial sets and the horizontal operators commute with the vertical ones. Note that, on the one hand, any bisimplicial set S provides two simplicial objects in the category of simplicial sets: the horizontal one \(S^h:\Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {SSet}\), \([p]\mapsto S_{p,\bullet }\), and the vertical one \(S^v:\Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {SSet}\), \([q]\mapsto S_{\bullet ,q}\). Then, by taking geometric realization, S gives rise to two simplicial spaces \([p]\mapsto S_{p,\bullet }\) and \([q]\mapsto S_{\bullet ,q}\). On the other hand, the bisimplicial set S also provides another simplicial set \(\mathrm {Diag}S:[p]\mapsto S_{p,p}\), whose face and degeneracy operators are given in terms of those of S by the formulas \(d_i=d_i^hd_i^v\) and \(s_i=s_i^hs_i^v\), respectively. It is known (e.g. [29, Lemma in page 86]) that there are natural homeomorphisms
and we usually take the geometric realization \(\!S\!\) of the bisimplicial set S to be
The following relevant fact is used several times along the paper (see [7, Chapter XII, 4.2 and 4.3] or [22, IV, Proposition 1.7], for example):
Fact 2.1
If \(f:S\rightarrow S^\prime \) is a bisimplicial map such that the maps \( f_{p,\bullet }: S_{p,\bullet }\rightarrow S^\prime _{p,\bullet }\) are homotopy equivalences for all p, then so is the map \(\mathrm {Diag}f: \mathrm {Diag}S \rightarrow \mathrm {Diag}S^\prime  \).
We will use also the \(\overline{W}\)construction on a bisimplicial set by Artin and Mazur [2, Sect. III], also called its “codiagonal” or “total complex”. Recall that, by viewing a bisimplicial set \(S:\Delta ^{\!{\mathrm {op}}}\times \Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {Set}\) as a horizontal simplicial object in the category of vertical simplicial sets, \(S:\Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {SSet}\), then the set of nsimplices of \(\overline{W}S\) is
and, for \(0\le i\le n\), the faces and degeneracies of an nsimplex are given by
There is a natural AlexanderWhitneytype diagonal approximation \(\mathrm {Diag}\,S\rightarrow \overline{W}S\),
inducing a homotopy equivalence on geometric realizations (see [16, 32, 36] for a proof).
2.2 Classifying spaces of 2categories
In Quillen’s development of Ktheory [29], the higher Kgroups are defined as the homotopy groups of a classifying space C associated to a (small) category C. This space is defined as
the geometric realization of the simplicial set termed its nerve
whose psimplices are length p sequences of composable morphisms in C (\({\mathrm {N}}_0 C=\mathrm {Ob}C\)).
The classifying space \(\!S\!\) of a simplicial category (that is, a simplicial object in \(\mathbf {Cat}\)) \(S:\Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {Cat}\) is defined as the geometric realization of the bisimplicial set \({\mathrm {N}}S:\Delta ^{\!{\mathrm {op}}}\times \Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {Set}\), \(([p],[q])\mapsto {\mathrm {N}}_qS_p \), obtained by composing S with the nerve functor \({\mathrm {N}}: \mathbf {Cat}\rightarrow \mathbf {SSet}\) from categories to simplicial sets. Thus, by (1), there is a natural homeomorphism
The notion of classifying space of a simplicial category provides the usual definition of the classifying space of a 2category. Although for the general background on 2categories used in this paper we refer to [6, 33], to fix some notation and terminology, we shall recall that a 2category C is just a category enriched in the category of small categories. Then, C is a category in which the homset between any two objects \(c,c'\in C\) is the set of objects of a category \(C(c,c')\), whose objects \(p:c\rightarrow c'\) are called 1cells and whose arrows are called 2cells and are denoted by \(\alpha :p\Rightarrow p'\) and depicted as
Composition in each category \(C(c,c')\), usually referred to as the vertical composition of 2cells, is denoted by \(\alpha \cdot \beta \). Moreover, the horizontal composition is a functor
that is associative and has identities \(1_c\in \mathcal {C}(c,c)\).
For any 2category C, the nerve construction (4) on it works by giving a simplicial category \({\mathrm {N}}C: \Delta ^{\mathrm {op}}\rightarrow \mathbf {Cat}\), whose classifying space is then the classifying space \(\!C\!\) of the 2category. Thus,
(the double bar notation \(\!\) avoids confusion with the classifying space C of the underlying category), where the bisimplicial set \({\mathrm {N}}{\mathrm {N}}C:\Delta ^{\!{\mathrm {op}}}\times \Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {Set}\), \(([p],[q])\mapsto {\mathrm {N}}_q{\mathrm {N}}_p C\), is the double nerve of C.
The following fact will be used.
Fact 2.2
([14, Lemma 2.6]) If two 2functors between 2categories \(F,G:A\rightarrow C\) are related by a lax or oplax transformation, \(F\Rightarrow G\), then there is an induced homotopy, \(\!F\!\Rightarrow \!G\!\), between the induced maps on classifying spaces \(\!F\!,\!G\!:\!A\!\rightarrow \!C \!\).
The category \(\mathbf {2}\mathbf {Cat}\) has a Thomasontype model structure, such that the classifying space functor \(C \mapsto \!C\!\) is an equivalence of homotopy theories between 2categories and topological spaces ([35, Theorem 4.5.1], [1, Théorème 6.27]). Hereafter, we will always consider \(\mathbf {2}\mathbf {Cat}\) with the Thomason model structure on it. Thus, for example, a 2functor \(F:A\rightarrow C\) is a weak equivalence if and only if the induced map \(\!F\!:\!A\!\rightarrow \!C\!\) is a homotopy equivalence, and a commutative square of 2categories and 2functors
is a homotopy pullback if and only if the induced square on classifying spaces
is a homotopy pullback of spaces. Later, we shall use basic properties of homotopy pullbacks of spaces, such as the two out of three property, etc. (see [17, Sect. 5] for instance). In particular, the homotopyfibre characterization [17, Sect. 5(2)], which easily leads us to assert that the square of 2categories above is a homotopy pullback whenever, for any object \(a\in A\), there is a commutative diagram of 2categories and 2functors
such that \(a\in \mathrm {Im}F\), both the left square and the composite square are homotopy pullbacks, and \(A'\) is weakly contractible ^{Footnote 3} in the sense that the functor from \(A'\) to the terminal (only one 2cell) 2category \(A'\rightarrow \star \) is a weak equivalence, that is, if the classifying space \(\!A'\!\) is contractible.
To conclude this preliminary section, we recall that the classifying space \(\!\!S\!\!\) of a simplicial 2category \(S:\Delta ^{\!{\mathrm {op}}}\rightarrow \mathbf {2}\mathbf {Cat}\), is the geometric realization of the simplicial space obtained by composing S with the classifying space functor \(\!\,\text {}\,\!:\mathbf {2}\mathbf {Cat}\rightarrow \mathbf {Top}\). Therefore,
where \(\mathrm {Diag}{\mathrm {N}}{\mathrm {N}}S\) is the simplicial set, \([p]\mapsto {\mathrm {N}}_p{\mathrm {N}}_pS_p\), diagonal of the trisimplicial set \({\mathrm {N}}{\mathrm {N}}S\).
2.3 The Grothendieck construction on 2functors
The socalled Grothendieck construction on diagrams of small categories underlies the 2categorical construction we treat here for 2diagrams of 2categories. For a more general version of the Grothendieck construction below, which works even on lax bidiagrams of bicategories (in the sense of Benabou), we refer the reader to [4, 9, 11–13].
We start by fixing some notations. Throughout the paper, the 2category of (small) 2categories, 2functors, and 2natural transformations is denoted by \({\underline{\mathbf{2Cat}}}\) (whereas \(\mathbf {2Cat}\), recall, denotes its underlying category of 2categories and 2functors). We view any category as a 2category in which all its 2cells are identities, and thus \({\underline{\mathbf{Cat}}}\) \(\subseteq \) \({\underline{\mathbf{2Cat}}}\) is the 2subcategory consisting of categories, functors, and natural transformations.
Further, if C is a 2category, the effect on cells of any 2functor \(\mathcal {D}:C \rightarrow \) \({\underline{\mathbf{2Cat}}}\) is denoted by
or, if \(\mathcal {D}:C^{\mathrm {op}} \rightarrow \) \({\underline{\mathbf{2Cat}}}\) is contravariant, by
Let C be a 2category, and let \(\mathcal {D}:C \rightarrow \) \({\underline{\mathbf{2Cat}}}\) be a 2functor. The Grothendieck construction on the 2diagram \(\mathcal {D}\) assembles the 2diagram into a 2category, denoted by
whose objects are pairs (a, x) with \(a\in \mathrm {Ob}C\) and \(x\in \mathrm {Ob}\mathcal {D}_{\!a}\), the 1cells are pairs \((f,u):(a,x)\rightarrow (b,y)\), where \(f:a\rightarrow b\) is a 1cell in C and \(u:f_*x\rightarrow y\) is a 1cell in \(\mathcal {D}_b\), and the 2cells
are pairs consisting of a 2cell of C together with a 2cell \(\phi :u\Rightarrow v\circ \alpha _*x\) in \(\mathcal {D}_b\),
The vertical composition of 2cells
is the 2cell
and the identity 2cell of a 1cell (f, u) as above is \(1_{(f,u)}=(1_f,1_u)\).
The horizontal composition of two 1cells is the 1cell
the identity 1cell of an object (a, x) is \(1_{(a,x)}=(1_a,1_x)\), and the horizontal composition of 2cells
is the 2cell
We denote by
the canonical projection 2functor given on cells by forgetting the second components
Note that a 2transformation \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) between 2functors \(\mathcal {D},\mathcal {E}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) induces the 2functor \(\int _C\Gamma : \int _C\mathcal {D}\rightarrow \int _C\mathcal {E}\) such that
Also, for \(\Gamma ':\mathcal {D}\Rightarrow \mathcal {E}\) any other 2transformation, a modification \(m:\Gamma \Rrightarrow \Gamma '\) gives rise to the 2transformation \( \int _C m:\int _C \Gamma \Rightarrow \int _C \Gamma ' \) given by
Thus, the 2categorical Grothendieck construction provides a 2functor
In a similar way, if \(\mathcal {D}:C^{\mathrm {op}} \rightarrow \) \({\underline{\mathbf{2Cat}}}\) is a 2functor, the Grothendieck construction on \(\mathcal {D}\) is the 2category, denoted by , whose objects are pairs (a, x) with \(a\in \mathrm {Ob}C\) and \(x\in \mathrm {Ob}\mathcal {D}_{\!a}\), whose 1cells \((f,u):(a,x)\rightarrow (b,y)\) are pairs where \(f:a\rightarrow b\) is a 1cell in C and \(u:x\rightarrow f^*y\) is a 1cell in \(\mathcal {D}_a\), and whose 2cells \((\alpha ,\phi ):(u,f)\Rightarrow (v,g)\) are pairs consisting of a 2cell \(\alpha :u\Rightarrow v\) of C together with a 2cell \(\phi :\alpha ^*y\circ u\Rightarrow v\) in \(\mathcal {D}_b\). As for the covariant case, the assignment \(\mathcal {D}\mapsto \int _C\mathcal {D}\) is the function on objects of a 2functor
2.4 The homotopyfibre 2functors
To finish this preliminary section, we review some needed results concerning the more striking examples of 2diagrams of 2categories: the 2diagrams of homotopyfibre 2categories of a 2functor.
As usual, if \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (resp. \(\mathcal {D}:C^{\mathrm {op}} \rightarrow \) \({\underline{\mathbf{2Cat}}}\)) and \(F:A\rightarrow C\) are 2functors, we denote by
(resp. \(F^*\mathcal {D}=\mathcal {D}\,F:A^{\mathrm {op}} \rightarrow \) \({\underline{\mathbf{2Cat}}}\)) the 2functor obtained by composing \(\mathcal {D}\) with F.
Let \(F:A\rightarrow C\) be any given 2functor. Then, for any object \(c\in C\), the homotopyfibre of F over c [14, 23], denoted by \(F\!\downarrow \!c\), is the 2category obtained by applying the Grothendieck construction on the 2functor \(F^*C(,c):A^{\mathrm {op}}\rightarrow {\underline{\mathbf{Cat}}}\), where \(C(,c):C^{\mathrm {op}}\rightarrow {\underline{\mathbf{Cat}}}\) is the hom 2functor; that is,
Thus, \(F\!\downarrow \!c\) has objects the pairs (a, p), with a a 0cell of A and \(p:Fa\rightarrow c\) a 1cell of C. A 1cell \((u,\phi ):(a,p)\rightarrow (a',p')\) consists of a 1cell \(u:a\rightarrow a'\) in A, together with a 2cell \(\phi :p\Rightarrow p'\circ Fu\) in the 2category C,
and, for \((u,\phi ),(u',\phi '):(a,p)\rightarrow (a',p')\), a 2cell \(\alpha :(u,\phi )\Rightarrow (u',\phi ')\) is a 2cell \(\alpha :u\Rightarrow u'\) in A such that \((1_{p'}\circ F\alpha )\cdot \phi =\phi '\). Compositions and identities are given canonically.
Any 1cell \(h:c\rightarrow c'\) in C gives rise to a 2functor
which acts on cells by
and, for \(h,h':c\rightarrow c'\), any 2cell \(\psi :h\Rightarrow h'\) in C produces a 2transformation
whose component at any object (a, p) of \(F\!\downarrow \!c\) is the 1cell of \(F\!\downarrow \!c'\)
In this way, we have the homotopyfibre 2functor
where \(\mathcal {Y}\) is the 2categorical Yoneda embedding; and, quite similarly, we also have the homotopyfibre 2functor
which assigns to each object c of C the homotopyfibre 2category of F under c, \(c\!\downarrow \!F\), whose objects are pairs \((a,c\overset{p}{\rightarrow }Fa)\). The 1cells \((u,\phi ):(a,p)\rightarrow (a',p')\) are pairs where \(u:a\rightarrow a'\) is a 1cell of A and \(\phi :Fu\circ p\Rightarrow p'\) is a 2cell of C, and a 2cell \(\alpha :(u,\phi )\Rightarrow (u',\phi ')\) is a 2cell \(\alpha :u\Rightarrow u'\) in A such that \(\phi '\cdot (F\alpha \circ 1_p)=\phi \).
In the particular case where \(F=1_C\) is the identity 2functor on C, we have the comma 2categories \(C\!\downarrow \!c\), of objects over an object c, and \(c\!\downarrow \!C\), of objects under c. The following fact is proved in [10, Theorem 4.1]:
Lemma 2.1
For any 2category C and any object \(c\in C\), the 2categories \(c\!\downarrow \!C\) and \(C\!\downarrow \!c\) are weakly contractible.
Returning to an arbitrary 2functor \(F:A\rightarrow C\), the 2diagrams \(F\!\downarrow \!\) and \(\!\downarrow \!F \) are relevant for homotopy interests, mainly because Quillen’s Theorem B in [29] has also been generalized for 2functors between 2categories as in Theorem 2.3 below. We shall first set some terminology. Following Dwyer, Kan, and Smith in [21, Sect. 6] and Barwick and Kan in [5], we say that:

F has the property B\(_l\) if, for any 1cell \(h:c\rightarrow c'\) in C, the induced 2functor \(h_*:F\!\downarrow \! c \rightarrow F\!\downarrow \! c'\) is a weak equivalence,^{Footnote 4}

F has the property B\(_r\) if, for any 1cell \(h:c\rightarrow c'\) in C, the induced 2functor \(h^*:c'\!\downarrow \! F \rightarrow c\!\downarrow \!F\) is a weak equivalence.
Now, observe that, for any object c of C, there are two pullback squares in \({\mathbf {2}\mathbf {Cat}}\)
where the vertical 2functors \(\pi \) are the canonical projections (9), and both 2functors \(\bar{F}\) act on cells by
Then, the extension of Quillen’s Theorem B for 2functors in [14, Theorem 3.2] tells us that the following theorem holds.
Theorem 2.3
A 2functor \(F : A\rightarrow C\) has the property \(\mathrm {B}_l\) (resp. \(\mathrm {B}_r\)) if and only if the left (resp. right) square in (10) is a homotopy pullback for every object \(c\in C\).
As an interesting consequence, we have the 2categorical version of Quillen’s Theorem A in [29] in Theorem 2.4 below. First, let us borrow some terminology from Hirschhorn in [26, 19.6.1]:

a 2functor \(F:A\rightarrow C\) is called homotopy left cofinal if all the homotopyfibre 2categories \(F\!\downarrow \!c \), \(c\in C\), are weakly contractible,^{Footnote 5}

a 2functor \(F:A\rightarrow C\) is called homotopy right cofinal if all the homotopyfibre 2categories \(c\!\downarrow \!F\), \(c\in C\), are weakly contractible.
Since every homotopy left (resp. right) cofinal 2functor has the property \(\mathrm {B}_l\) (resp. \(\mathrm {B}_r\)), from Theorem 2.3, it follows the following extension of Quillen’s Theorem A, which was originally proved in [10, Theorem 1.2].
Theorem 2.4
Every homotopy left or right cofinal 2functor between 2categories is a weak equivalence.
3 The homotopy colimit and homotopy invariance theorems
As we said in the introduction, the goal of this paper is to state and prove the more relevant homotopy properties of the Grothendieck construction on 2diagrams of 2categories. Most of this section is dedicated to proving Theorem 3.1 below, which has Thomason’s homotopy colimit theorem [34, Theorem 1.2] as its natural precedent and also includes the results in [14, Theorem 4.5] as particular cases. The result therein, on which many further results are based, states that, for any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the classifying space of the 2category \(\int _{C}\mathcal {D}\) can be realized, up to homotopy equivalence, through the simplicial bar construction on \(\mathcal {D}\), also called Borel construction, or simplicial replacement, denoted by \(\mathcal {B}_C\mathcal {D}\) and defined as follows.^{Footnote 6}
Definition 3.1
Let C be a 2category and \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) be a 2functor. The simplicial bar construction on \(\mathcal {D}\) is the simplicial 2category
whose 2category of psimplices is
and whose face and degeneracy 2functors are defined as follows: The face 2functor \(d_0\) is induced by the 2functor
which carries an object \((x,c_0\overset{f}{\rightarrow }c_1)\) to the object \(f_*x\). A 1cell \((u,\alpha ):(x,f)\rightarrow (y,g)\) is carried by \(d_0\) to the composite 1cell \(g_*u\circ \alpha _*x:f_*x\rightarrow g_*y\),
and \(d_0\) acts on 2cells by
The other face and degeneracy 2functors are induced by the operators \(d_i\) and \(s_i\) in \({\mathrm {N}}C\) as \(1_{\!\mathcal {D}_{\!c_0}}\!\times d_i\) and \(1_{\!\mathcal {D}_{\!c_0}}\!\times s_i\), respectively.
It is not hard to see that, if \(\mathcal {D},\mathcal {E}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) are 2functors, then any 2transformation \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) gives rise to a simplicial functor \(\Gamma _*:\mathcal {B}_C\mathcal {D}\rightarrow \mathcal {B}_C\mathcal {E}\); and also that a modification \(m:\Gamma \Rrightarrow \Gamma '\), where \(\Gamma ':\mathcal {D}\Rightarrow \mathcal {E}\) is any other 2transformation, induces a simplicial transformation \(m_*:\Gamma _*\Rightarrow \Gamma '_*\). Thus, the simplicial bar construction provides a 2functor
There is an analogous construction \(\mathcal {B}_C\mathcal {D}\) for contravariant 2functors \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\). This is as follows:^{Footnote 7} We call the simplicial bar construction on \(\mathcal {D}\) the simplicial 2category
whose 2category of psimplices is
and whose faces and degeneracies are induced by the corresponding ones in \({\mathrm {N}}C\), as \(d_i\times 1_{\!\mathcal {D}_{\!c_p}}\) and \(s_i\times 1_{\!\mathcal {D}_{\!c_p}}\), for \(0\le i<p\), whereas the face 2functor \(d_p\) is induced by the 2functor
which acts on cells by
Thus, the construction \(\mathcal {D}\mapsto \mathcal {B}_C\mathcal {D}\) is the function on objects of a 2functor
Theorem 3.1
(The homotopy colimit theorem) For any 2functor \(\mathcal {D}: C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (or \(\mathcal {D}: C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)), where C is a 2category, there exists a natural homotopy equivalence
Proof
We shall treat only the covariant case, as the other is proven similarly. The strategy of the proof is as follows: For any given 2functor \(\mathcal {D}: C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), we build a trisimplicial set \(E=E(\mathcal {D})\), together with isomorphisms of simplicial sets
so that, by Fact 2.1 and (3), we have a homotopy equivalence
As \( \!\int _{C}\mathcal {D}\!=\mathrm {Diag}{\mathrm {N}}{\mathrm {N}}\int _{C}\mathcal {D}\overset{(3)}{\simeq }{\overline{W}}{\mathrm {N}}{\mathrm {N}}\int _{C}\mathcal {D}\), while
the proof will be complete.
Before starting the construction of E, we shall describe, for any 2category, C say, the simplicial set \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\). To do that, let us first represent a (p, q)simplex of the bisimplicial set \({\mathrm {N}}{\mathrm {N}}C\) as a diagram \((c,f,\alpha )_{p,q}\) in C of the form
whose horizontal iface is obtained by deleting the object \(c_{q+i}\) and using, for \(0<i<p\), the composite cells \(f^k_{q+i+1}\circ f^k_{q+i}\) and \(\alpha ^k_{q+i+1}\circ \alpha ^k_{q+i}\) to rebuild the new \((p1,q)\)simplex, and whose vertical jface is obtained by deleting all the 1cells \(f^j_{q+m}\) and using the composite 2cells \(\alpha ^{j+1}_{q+m}\cdot \alpha ^{j}_{q+m}\), for \(0<j<q\), to complete the \((p,q1)\) simplex of \({\mathrm {N}}{\mathrm {N}}B\). Then, it is straightforward to obtain the following description of the simplicial set \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\): The vertices are the objects \(c_0\) of C, the 1simplices are the 1cells \(f_1^0:c_0\rightarrow c_1\) of C, and, for \(n\ge 2\), the nsimplices are diagrams \((c,f,\alpha )_n\) in C of the form
that is, they consist of objects \(c_m\) of C, \(0\le m\le n\), 1cells \(f_m^k:c_{m1}\rightarrow c_{m}\), \(0\le k <m\le n\), and 2cells \(\alpha _m^k:f_m^{k1}\Rightarrow f_m^{k}\), \(0<k<m\le n\). The simplicial operators of \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\) act much as for the usual nerve of an ordinary category: The iface of an nsimplex as in (16) is obtained by deleting the object \(c_i\) and the 1cells \(f_m^{i}:c_{m1}\rightarrow c_m\), for \(i< m\), and using the composite 1cells \(f^k_{i+1}\circ f^k_i:c_{i1}\rightarrow c_{i+1}\), \( k<i\), the horizontally composite 2cells \(\alpha ^k_{i+1}\circ \alpha ^k_i:f^{k1}_{i+1}\circ f^{k1}_i\Rightarrow f^k_{i+1}\circ f^k_i\), \(0<k<i\), and the vertically composed 2cells \(\alpha ^{i+1}_m\cdot \alpha ^i_m:f^{i1}_m\Rightarrow f^{i+1}_m\), when \(i<m1\), to complete the new \((n1)\)simplex. The idegeneracy of \((c,f,\alpha )_n\) is constructed by repeating the object \(c_i\) at the \(i + 1\)place and inserting \(i+1\) times the identity 1cell \(1_{c_i}:c_i\rightarrow c_i\), i times the identity 2cell \(1_{1_{c_i}}:1_{c_i}\Rightarrow 1_{c_i}\) and, for each \(i<m\), by replacing the 1cell \(f_m^i:c_{m1}\rightarrow c_m\) by the identity 2cell \(1_{f_m^i}:f_m^i\Rightarrow f_m^i\).
Then, the simplicial set \({\overline{W}}([p]\mapsto {\overline{W}}{\mathrm {N}}{\mathrm {N}}\mathcal {B}_C\mathcal {D}_p)\) can be described as follows: Its nsimplices are pairs
where \((c,f,\alpha )_n\) is an nsimplex of \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\) as in (16), whereas \((x,u,\phi )_n\) is a list with a diagram in each 2category \(\mathcal {D}_{\!c_0}\),...,\(\mathcal {D}_{\!c_n}\) of the form
That is, \((x,u,\phi )_n\) consists of 0cells \(x_k\) of \(\mathcal {D}_{\!c_k}\), \(0\le k\le n\), 1cells \(u_m^k:f^{m1}_{m*}x_{m1}\rightarrow x_{m}\), \(0\le k <m\le n\), and 2cells \(\phi _m^k:u_m^{k1}\Rightarrow u_m^{k}\), \(0<k<m\le n\). Further, the iface of the simplex \(((c,f,\alpha )_n,(x,u,\phi )_n)\) is obtained by taking the iface of \((c,f,\alpha )_n\) in the simplicial set \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\) and, in a similar way, by deleting the object \(x_i\) and the 1cells \(u_m^{i}:f_{m*}^{m1}x_{m1}\rightarrow x_m\), for \(i< m\), and then using the composite 1cells
for \(k<i\), the horizontally composite 2cells
for \(0<k<i\), and the vertically composed 2cells \(\phi ^{i+1}_m\cdot \phi ^i_m:u^{i1}_m\Rightarrow u^{i+1}_m\), \(i<m1\), to complete the new \((n1)\)simplex. Similarly, the idegeneracy of \(((c,f,\alpha )_n,(x,u,\phi )_n)\) is given by first taking the idegeneracy of \((c,f,\alpha )_n\) in the simplicial set \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\) and second by repeating the object \(x_i\), inserting \(i+1\) times the identity 1cell \(1_{x_i}:x_i\rightarrow x_i\), i times the identity 2cell \(1_{1_{x_i}}:1_{x_i}\Rightarrow 1_{x_i}\) and, for each \(i<m\), by replacing the 1cell \(u_m^{i}\) by the identity 2cell \(1_{u_m^i}:u_m^i\Rightarrow u_m^i\).
Now we construct \(E=E(\mathcal {D})\) as the trisimplicial set whose (p, n, q)simplices are pairs
with \((c,f,\alpha )_{p,q}\) a (p, q)simplex of \({\mathrm {N}}{\mathrm {N}}C\), as in (15), and \((x,u,\phi )_{p,n,q}\) a system of data consisting of a diagram in each 2category \(\mathcal {D}_{\!c_{q+1}}\), ..., \(\mathcal {D}_{\!c_{q+p}}\) of the form
That is, it consists of objects \(x_{q}\in D_{\!c_q}\), \(\ldots \), \(x_{q+p}\in \mathcal {D}_{\!c_{q+p}}\), 1cells \(u^k_{q+m}:f^q_{q+m*}x_{q+m1}\rightarrow x_{q+m}\), \(0\le k\le n\), \(0< m\le p\), and 2cells \(\phi ^k_{q+m}:u^{k1}_{q+m}\Rightarrow u^{k}_{q+m}\), \(0< k\le n\), \(0< m\le p\).
The iface in the pdirection map of E carries the (p, n, q)simplex (20) to the \((p1,n,q)\)simplex obtained by taking the horizontal iface of \((c,f,\alpha )_{p,q}\) in \({\mathrm {N}}{\mathrm {N}}C\), deleting the object \(x_{q+i}\), and using the composite 1cells
and the horizontally composite 2cells
to complete the new \((p1,n,q)\)simplex.
The jface in the ndirection of the (p, n, q)simplex (20) is obtained by keeping \((c,f,\alpha )_{p,q}\) unaltered, deleting all the 1cells \(u^j_{q+m}\), and using, when \(0<j<n\), the composite 2cells \(\phi ^{j+1}_{q+m}\cdot \phi ^{j}_{q+m}\) to complete the simplex.
For any \(k<q\), the kface in the qdirection of the (p, n, q)simplex (20) is given by replacing \((c,f,\alpha )_{p,q}\) by its vertical kface in \({\mathrm {N}}{\mathrm {N}}C\) and keeping \((x,u,\phi )_{p,n,q}\) unchanged, while its qface consists of the vertical qface of \((c,f,\alpha )_{p,q}\) in \({\mathrm {N}}{\mathrm {N}}C\) (which, recall, is obtained by deleting the 1cells \(f^q_{q+m}\)) together with the list of diagrams
With degeneracies given in a standard way, it is straightforward to see that E is a trisimplicial set. Then, an easy verification shows that the isomorphisms in (13),
holds.
Now, an analysis of the simplicial set \({\overline{W}}E_{p,\bullet ,\bullet }\) says that its qsimplices are pairs
with a (p, q)simplex \((c,f,\alpha )_{p,q}\) of \({\mathrm {N}}{\mathrm {N}}C\), as in (15), together with data \((x,u,\phi )_{p,q}\) consisting of a diagram in each 2category \(\mathcal {D}_{\!c_{q+1}}\), ..., \(\mathcal {D}_{\!c_{q+p}}\) of the form
More precisely, \((x,u,\phi )_{p,q}\) consists of objects
1cells
and 2cells
The jface of such a qsimplex (20) is given by taking the vertical jface of \((c,f,\alpha )_{p,q}\) in \({\mathrm {N}}{\mathrm {N}}C\), deleting the 1cells \(u^j_{q+m}\), and inserting the pasted 2cells below, for \(0<j<q\).
Then, an easy and straightforward verification shows that an nsimplex of the simplicial set \({\overline{W}}([p]\mapsto {\overline{W}}E_{p,\bullet ,\bullet })\) is a pair
where \((c,f,\alpha )_n\) is an nsimplex of \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\) as in (16), while \((x,u,\phi )_n\) is a list with a diagram in each 2category \(\mathcal {D}_{\!c_0}\),...,\(\mathcal {D}_{\!c_n}\) of the form
That is, \((x,u,\phi )_n\) consists of 0cells
1cells
and 2cells
Further, the iface of the simplex \(((c,f,\alpha )_n,(x,u,\phi )_n)\) is obtained by taking the iface of \((c,f,\alpha )_n\) in the simplicial set \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\) and, in a similar way, by deleting the object \(x_i\) and all the 1cells \(u_m^{i}:f_{m*}^{i}x_{m1}\rightarrow x_m\), for \(i< m\), and then using the composite 1cells
for \(k<i\), and the pasted 2cells
for \(0<k<i\), and
for \(0<i<m1\), to complete the iface \((n1)\) simplex. Similarly, the idegeneracy of \(((c,f,\alpha )_n,(x,u,\phi )_n)\) is given by first taking the idegeneracy of \((c,f,\alpha )_n\) in the simplicial set \({\overline{W}}{\mathrm {N}}{\mathrm {N}}C\) and secondly by repeating the object \(x_i\), inserting \(i+1\) times the identity 1cell \(1_{x_i}:x_i\rightarrow x_i\), i times the identity 2cell \(1_{1_{x_i}}:1_{x_i}\Rightarrow 1_{x_i}\) and, for each \(i<m\), by repeating the 1cell \(u_m^{i}:f^i_{m*}x_{i1}\rightarrow x_i\) and inserting the identity 2cell \(1_{u_m^i}:u_m^i\Rightarrow u_m^i\).
Finally, observe that any nsimplex \(((c,f,\alpha )_n,(x,u,\phi )_n)\) of \({\overline{W}}([p]\mapsto {\overline{W}}E_{p,\bullet ,\bullet })\), such as (21), identifies with the nsimplex \(((c,x),(f,u),(\alpha ,\phi ))_n \in {\overline{W}}{\mathrm {N}}{\mathrm {N}}\int _{C}\mathcal {D}\),
given by the objects \((c_m,x_m)\) of \(\int _C\mathcal {D}\), \(0\le m\le n\), the 1cells
and 2cells
Thus, the claimed simplicial isomorphism in (14) holds. \(\square \)
The first basic property below quickly follows from Theorem 3.1.
Theorem 3.2
(Homotopy Invariance Theorem) Let \(\mathcal {D},\mathcal {E}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (or \(\mathcal {D},\mathcal {E}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) be 2functors, where C is any 2category. If \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) is a 2transformation such that, for each object c of C, the 2functor \(\Gamma _c:\mathcal {D}_c\rightarrow \mathcal {E}_c\) is a weak equivalence of 2categories, then the induced map on classifying spaces
is a homotopy equivalence.
Proof
By Theorem 3.1, we can argue with \(\mathcal {B}_C\) instead of \(\int _\mathcal {C}\). Since \(\Gamma \) is objectwise a weak equivalence, for any integer \(p\ge 0\) the induced 2functor \(\mathcal {B}_C\mathcal {D}_p\rightarrow \mathcal {B}_C\mathcal {E}_p\),
is a weak equivalence. Then, by Fact 2.1, \(\!\!\mathcal {B}_C\mathcal {D}\!\!=[p]\mapsto \!\mathcal {B}_C\mathcal {D}_p\! \simeq [p]\mapsto \!\mathcal {B}_C\mathcal {E}_p\!= \!\!\mathcal {B}_C\mathcal {E}\!\!\). \(\square \)
If \(\mathrm {Ct}D:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) denote the constant 2functor on a 2category C given by a 2category D, then the Grothendieck construction on it just gives the product 2category of C and D, that is,
In particular, for \(D=\star \) the terminal 2category, \( \int _C\,\mathrm {Ct}\star =C\times \star \cong C\). We use this elemental observation in the proof of the following corollary.
Corollary 3.1
Let C be a 2category and \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (or \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) a 2functor such that, for any object c of C, the 2category \(\mathcal {D}_{\!c}\) is weakly contractible. Then, the projection 2functor (9), \(\pi : \int _C\mathcal {D}\rightarrow C\), is a weak equivalence.
Proof
By Theorem 3.2, the induced 2functor by the collapse 2transformation \(\mathcal {D}\Rightarrow \mathrm {Ct}\star \), \(\int _C\mathcal {D}\rightarrow \int _C\,\mathrm {Ct}\star \) is a weak equivalence. As the projection 2functor \(\pi \) is the composite \(\int _C\mathcal {D}\rightarrow \int _C\,\mathrm {Ct}\star =C\times \star \cong C\), the result follows. \(\square \)
4 Theorems A and B
In this section, we state and prove extensions of Theorems 2.3 and 2.4 for 2transformations between 2diagrams of 2categories.
For any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (or \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) and any object \(c\in C\), let
denote the embedding 2functor
If \(\mathcal {D},\,\mathcal {E}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) are 2functors and \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) is a 2transformation, for any objects \(c\in C\) and \(y\in \mathcal {E}_c\), there is a canonical commutative square of 2categories
which, keeping the notations in squares (10), is the composite of the squares
and we have the theorem below.
Theorem 4.1
(Theorem \(\mathrm {B}_l\) for 2transformations) Let \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) be a 2transformation, where \(\mathcal {D},\,\mathcal {E}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) are 2functors. The following properties are equivalent:

(a)
For any object c in C and any object y in \(\mathcal {E}_c\), the square (22) is a homotopy pullback.

(b)
The 2functor \(\int _C\Gamma :\int _C\mathcal {D}\rightarrow \int _C\mathcal {E}\) has the property \(\mathrm {B}_l\).

(c)
The two conditions below hold.

\(\mathrm {B1}_l\): For each object c of C, the 2functor \(\Gamma _c:\mathcal {D}_c\rightarrow \mathcal {E}_c\) has the property \(\mathrm {B}_l\).

\(\mathrm {B2}_l\): For each 1cell \(h:c\rightarrow c'\) in C and any object \(y\in \mathcal {E}_c\), the induced 2functor
is a weak equivalence.


(d)
The two conditions below hold.

\(\mathrm {B1}_l\) For each object c of C, the 2functor \(\Gamma _c:\mathcal {D}_c\rightarrow \mathcal {E}_c\) has the property \(\mathrm {B}_l\).

\(\mathrm {B2}'_l\) For each 1cell \(h:c\rightarrow c'\) in C, the square
(23)is a homotopy pullback.

Proof
In order to prove the result, first we show that, for any object c in C and any object y in \(\mathcal {E}_c\), there is a weak equivalence
Here, R is the 2functor acting on cells of the 2category \(\int _C\!\Gamma \! \downarrow \!(c,y)\) in the following way: On objects ((a, x), (p, v)), where a is an object of C, x an object of \(\mathcal {D}_a\), \(p:a\rightarrow c\) is a 1cell in C, and \(v:p_*\Gamma _{\!a}x=\Gamma _{\!c}p_*x\rightarrow y\) is a 1cell in \(\mathcal {E}_c\),
On 1cells
where \(f:a\rightarrow a'\) is in C, \(u:f_*x\rightarrow x'\) in \(\mathcal {D}_a\), \(\alpha :p\Rightarrow p'\circ f\) in C, and \(\psi :v\Rightarrow v'\circ p'_*\Gamma _{\!a'}u\circ \alpha _*\Gamma _{\!a}x\) in \(\mathcal {E}_c\),
And, for a 2cell \((\beta ,\phi ):((f,u),(\alpha ,\psi ))\Rightarrow ((f',u'),(\alpha ',\psi '))\), where \(\beta :f\Rightarrow f'\) is in C and \(\phi :u\Rightarrow u'\circ \beta _*x\) in \(\mathcal {D}_a\), satisfying the corresponding conditions,
This 2functor R is actually a retraction, with a section given by the induced 2functor on homotopyfibre 2categories
making the diagram below commutative:
Explicitly, \(\bar{c}\) in (26) acts on objects (x, v), where x is an object of \(\mathcal {D}_c\) and \(v:\Gamma _{\!c}x\rightarrow y\) is a 1cell of \(\mathcal {E}_c\), by
on 1cells \((u,\psi ):(x,v)\rightarrow (x',v')\), where \(u:x\rightarrow x'\) is in \(\mathcal {D}_c\) and \(\psi :v\Rightarrow v'\circ \Gamma _{\!c}u\) in \(\mathcal {E}_c\), by
and, on a 2cell \(\phi :(u,\psi )\Rightarrow (u',\psi ')\),
It is plain to see that \(R\,\bar{c}=1\). Furthermore, there is an oplax transformation \(1\Rightarrow \bar{c}\,R\) given, on any object ((a, x), (p, v)) of \(\int _C\!\Gamma \! \downarrow \!(c,y)\), by the 1cell
and whose naturality component at any 1cell as in (25) is
Hence, for the maps induced by R and \(\bar{c}\) on classifying spaces, we have \(\!R\!\,\!\bar{c}\!=1\) and, by Fact 2.2, \(1\simeq \!\bar{c}\!\,\!R\!\). Thus, it follows that both 2functors R and \(\bar{c}\) are weak equivalences.
Let us now observe that the square (22) is the composite of the squares
Therefore, as both vertical 2functors \(\bar{c}\) are weak equivalences, the square (22) is a homotopy pullback if and only if the square (I) above is as well. Thus, by Theorem 2.3, it follows that \((a)\Leftrightarrow (b)\).
To prove \((b)\Leftrightarrow (c)\), let us observe that, for any 1cell \((h,w):(c,y)\rightarrow (c',y')\) in \(\int _C\mathcal {E}\), there is a commutative diagram of 2functors
where both vertical 2functors R are weak equivalences. If the 2transformation \(\Gamma \) has the properties \(\mathrm {B1}_l\) and \(\mathrm {B2}_l\), then both 2functors \(\bar{h}_*\) and \(w_*\) in the bottom of the diagram above are weak equivalences, and therefore the 2functor \(\overline{(h,w)}_*\) at the top is also a weak equivalence. That is, the 2functor \(\int _C\Gamma \) has the property \(\mathrm {B}_l\).
Conversely, assume that \(\int _C\Gamma \) has the property \(\mathrm {B}_l\). Then, for any object c of C and any 1cell \(w:y\rightarrow y'\) of \(\mathcal {E}_c\), the above commutative square for the case where \(h=1_c\) proves that the 2functor \(w_*:\Gamma _{\!c}\!\downarrow \!y\rightarrow \Gamma _{\!c}\!\downarrow \!y'\) is a weak equivalence; that is, \(\Gamma _{\!c}:\mathcal {D}_c\rightarrow \mathcal {E}_c\) has the property \(\mathrm {B}_l\). Similarly, the commutativity of the above square for \(w=1_y\) implies that, for every 1cell \(h:c\rightarrow c'\) on C and any object \(y\in \mathcal {E}_c\), the 2functor \(\bar{h}_*:\Gamma _{\!c}\!\downarrow \!y\rightarrow \Gamma _{\!c'}\!\downarrow \!h_*y\) is a weak equivalence.
Finally, the equivalence \((c)\Leftrightarrow (d)\) is consequence of the homotopy fibre characterization of homotopy pullbacks of spaces (hence of 2categories, see Sect. 2.2): For any 1cell \(h:c\rightarrow c'\) in C and any object \(y\in \mathcal {E}_c\), we have the equality of composite squares \((I)+(II)=(III)+(IV)\), where
Under the hypothesis \(\mathrm {B1}_l\), the squares (I) and (IV) are both homotopy pullbacks (where, recall, the comma 2categories \(\mathcal {E}_{\!c}\!\downarrow \!y\) and \(\mathcal {E}_{\!c'}\!\downarrow \!h_*y\) are weakly contractible). Then, as the square \((II)=(23)\) is a homotopy pullback if and only if, for any object \(y\in \mathcal {E}_y\), the square \((I)+(II)=(III)+(IV)\) is a homotopy pullback, we conclude that the square (23) is a homotopy pullback if and only if the square (III) is as well, which holds if and only if the 2functor \(\bar{h}_*:\Gamma _{\!c}\!\downarrow \!y\rightarrow \Gamma _{\!c'}\!\downarrow \!h_*y\) is a weak equivalence. \(\square \)
Similarly, if \(\mathcal {D},\,\mathcal {E}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\) are 2functors and \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) is a 2transformation, for any objects \(c\in C\) and \(y\in \mathcal {E}_{c}\), there is a commutative square
defined as the composite of the squares
and we have the theorem below.
Theorem 4.2
(Theorem \(\mathrm {B}_r\) for 2transformations) Let \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) be a 2transformation, where \(\mathcal {D},\,\mathcal {E}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\) are 2functors. The following properties are equivalent:

(a)
For any object c in C and any object y in \(\mathcal {E}_c\), the square (27) is a homotopy pullback.

(b)
The 2functor \(\int _C\Gamma :\int _C\mathcal {D}\rightarrow \int _C\mathcal {E}\) has the property \(\mathrm {B}_r\).

(c)
The two conditions below hold.

\(\mathrm {B1}_r\): For each object c of C, the 2functor \(\Gamma _c:\mathcal {D}_c\rightarrow \mathcal {E}_c\) has the property \(\mathrm {B}_r\).

\(\mathrm {B2}_r\): For each 1cell \(h:c\rightarrow c'\) in C and any object \(y'\in \mathcal {E}_{c'}\), the induced 2functor
is a weak equivalence.


(d)
The two conditions below hold.

\(\mathrm {B1}_r\): For each object c of C, the 2functor \(\Gamma _c:\mathcal {D}_c\rightarrow \mathcal {E}_c\) has the property \(\mathrm {B}_r\).

\(\mathrm {B2}'_r\): For each 1cell \(h:c\rightarrow c'\) in C, the square
is a homotopy pullback.

Proof
This is parallel to the proof of Theorem 4.1 given above, and we leave it to the reader. We simply note that, in this case, the weak equivalence
for each objects \(c\in C\) and \(y\in \mathcal {E}_c\), is defined as below.
On objects \(((x,a),(p,v))\in (c,y)\! \downarrow \!\int _C\!\Gamma \), where a is an object of C, x an object of \(\mathcal {D}_a\), \(p:c\rightarrow a\) is a 1cell in C and \(v:y\rightarrow p^*\Gamma _{\!a}x=\Gamma _{\!c}p^*x\) is a 1cell in \(\mathcal {E}_c\),
On 1cells \(((f,u),(\alpha ,\psi )):((a,x),(p,v))\rightarrow ((a',x'),(p',v'))\) where \(f:a\rightarrow a'\) is in \(\mathcal {C}\), \(u:x\rightarrow f^*x'\) in \(\mathcal {D}_a\), \(\alpha : f\circ p\Rightarrow \circ p'\) in C, and \(\psi : \alpha ^*\Gamma _{\!a'}x'\circ p^*\Gamma _{\!a}u\circ v \Rightarrow v'\) in \(\mathcal {E}_c\),
and, for a 2cell \((\beta ,\phi ):((f,u),(\alpha ,\psi ))\Rightarrow ((f',u'),(\alpha ',\psi '))\), where \(\beta :f\Rightarrow f'\) is in C and \(\phi :\beta ^*x'\circ u\Rightarrow u'\) in \(\mathcal {D}_a\),
\(\square \)
Observe that, in the particular case where \(C=\star \) the terminal 2category, Theorems 4.1 and 4.2 state exactly the same as Theorem 2.3.
Furthermore, in the specific case where \(\mathcal {E}=\star \) is the constant terminal 2category, for any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) or \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the projection 2functors \(\pi \) are actually isomorphisms \(\Gamma _{\!c}\!\downarrow \!\star \cong \mathcal {D}_{\!c}\cong \star \!\downarrow \!\Gamma _{\!c}\), and Theorems 4.1 and 4.2 give as a corollary the following 2categorical version of the relevant Quillen’s detection principle for homotopy pullback diagrams [29, Lemma in p. 14] (see [11, Theorem 4.3] for a more general bicategorical result). Let us also stress that the weak equivalences (24) and (28), in this case where \(\mathcal {E}=\star \), establish weak equivalences
between the homotopyfibre 2categories of the projection 2functor \(\pi :\int _C\mathcal {D}\rightarrow C\) over the objects of C and the 2categories attached by the 2diagram to these objects.
Corollary 4.1
(Detecting homotopy pullbacks) Let C be a 2category. For any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (resp. \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)), the following statements are equivalent:

(a)
For any object c in C the square
(29)is a homotopy pullback.

(b)
The projection 2functor \(\pi :\int _C\mathcal {D}\rightarrow C\) has the property \(\mathrm {B}_l\) (resp. \(\mathrm {B}_r\)).

(c)
For each 1cell \(h:c\rightarrow c'\) of C, the 2functor \(h_*:\mathcal {D}_{\!c}\rightarrow \mathcal {D}_{\!c'}\) (resp. \(h^*:\mathcal {D}_{\!c'}\rightarrow \mathcal {D}_{\!c}\)) is a weak equivalence.
The following consequence of Theorems 4.1 and 4.2 is closely related to Theorems 3.2 and 2.4.
Corollary 4.2
(Theorem A for 2diagrams) Let \(\Gamma :\mathcal {D}\Rightarrow \mathcal {E}\) be a 2transformation, where \(\mathcal {D},\,\mathcal {E}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (resp. \(\mathcal {D},\,\mathcal {E}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) are 2functors. The following statements are equivalent:

(a)
The 2functor \(\int _C\Gamma :\int _C\mathcal {D}\rightarrow \int _C\mathcal {E}\) is homotopy left (resp. right) cofinal.

(b)
For any object \(c\in C\), the 2functor \(\Gamma _{\!c}:\mathcal {D}_c\rightarrow \mathcal {E}_c\) is homotopy left (resp. right) cofinal.
5 Changing the indexing 2category
If \(F:A\rightarrow C\) is a 2functor between 2categories, then any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), or \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\), gives rise to a pullback of 2categories
where \(\bar{F}\) is given by
Our first result here completes Corollary 4.1:
Theorem 5.1
Let C be a 2category and \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (resp. \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) a 2functor. The following statements are equivalent:

(a)
For any 2functor \(F:A\rightarrow C\), the square (30) is a homotopy pullback.

(b)
For any 1cell \(h:c\rightarrow c'\) in C, the 2functor \(h_*:\mathcal {D}_{\!c}\rightarrow \mathcal {D}_{\!c'}\) (resp. \(h^*:\mathcal {D}_{\!c'}\rightarrow \mathcal {D}_{\!c}\)) is a weak equivalence.
Proof
Suppose (a) holds. Let \(c:\mathrm { pt}\rightarrow C\) be the 2functor given by any object \(c\in C\). As we have quite an obvious isomorphism \(\int _{\star }c^*\mathcal {D}\cong \mathcal {D}_{\!c}\), the square
is, by hypothesis, a homotopy pullback. Hence, the result follows from Corollary 4.1.
Conversely, assume (b) holds. Again by Corollary 4.1, for any object \(c\in C\), the square (32) above is a homotopy pullback. Since, for any given 2functor \(F:A\rightarrow C\), the 2functor \(F^*\!\mathcal {D}:A\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (resp. \(F^*\!\mathcal {D}:A^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) trivially is under the same hypothesis (b) as \(\mathcal {D}\), it follows that, for any object \(a\in A\), both the left side and the composite square in the diagram
are homotopy pullbacks. Then, from the homotopy fibre characterization of homotopy pullbacks, it follows that the right side square above is a homotopy pullback, as required. \(\square \)
Next, we state the complementary counterpart to the theorem above. If \(F:A\rightarrow C\) and \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\) are 2functors, for any objects \(c\in C\) and \(z\in \mathcal {D}_{\!c}\), let
be the 2functor defined on cells by
Theorem 5.2
For a 2functor \(F:A\rightarrow C\), the following statements are equivalent:

(a)
\(F:A\rightarrow C\) has the property \(\mathrm {B}_l\).

(b)
For any 2functor \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the 2functor \(\bar{F}:\int _AF^*\mathcal {D}\rightarrow \int _C\mathcal {D}\) has the property \(\mathrm {B}_l\).

(c)
For any 2functor \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\), and any objects \(c\in C\) and \(z\in \mathcal {D}_{\!c}\), the commutative square
(34)is a homotopy pullback.

(d)
For any 2functor \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the square (30)
is a homotopy pullback.
Proof
For any objects \(c\in C\) and \(z\in \mathcal {D}_{\!c}\), let
be the induced 2functor on homotopyfibre 2categories making the diagram below commutative:
Explicitly, this \(\bar{\pi }\) acts on objects ((a, x), (p, v)), where a is an object of A, x an object of \(\mathcal {D}_{F\!a}\), \(p:Fa\rightarrow c\) is a 1cell in C and \(v:x\rightarrow p^*z\) is a 1cell in \(\mathcal {D}_{Fa}\), by
On 1cells
where \(f:a\rightarrow a'\) is a 1cell of A, \(u:x\rightarrow (Ff)^*x'\) of \(\mathcal {D}_{F\!a}\), \(\phi :p\Rightarrow p'\circ Ff\) a 2cell of C, and \(\beta :\phi ^*z\circ v\Rightarrow (Ff)^*v'\circ u\) a 2cell of \(\mathcal {D}_{F\!a}\),
and, on a 2cell \((\alpha ,\psi ):((f,u),(\phi ,\beta ))\Rightarrow ((f',u'),(\phi ',\beta '))\), where \(\alpha :f\Rightarrow f'\) is a 2cell of A and \(\psi :(F\alpha )^*x'\circ u\Rightarrow u'\) a 2cell in \(\mathcal {D}_{Fa}\), satisfying the corresponding conditions,
The 2functor \(\bar{\pi }\) is actually a retraction, with a section given by the 2functor
which acts on objects by
on a 1cell \((f,\phi ):(a,p)\rightarrow (a',p')\) by
and on a 2cell \(\alpha :(f,\phi )\Rightarrow (f',\phi ')\) by
It is clear that \(\bar{\pi }\,i_z=1\). Furthermore, there is an oplax transformation \(1\Rightarrow i_z\,\bar{\pi }\) given, on any object ((a, x), (p, v)) of \(\bar{F}\!\downarrow \!(c,z)\), by the 1cell
and whose naturality component at any 1cell as in (36) is
Hence, for the maps induced by \(\bar{\pi }\) and \(i_z\) on classifying spaces, \(\!\bar{\pi }\!\,\!i_z\!=1\) and, by Fact 2.2, \(1\simeq \!i_z\!\,\!\bar{\pi }\!\). Therefore, both 2functors \(\bar{\pi }\) and \(i_z\) are weak equivalences.
Let us now observe that the square (34) is the composite of the squares
where both vertical 2functors \(i_z\) are weak equivalences. It follows that the square (34) is a homotopy pullback if and only if the square (I) above is as well. As, by Theorem 2.3, the squares (I) are homotopy pullbacks, for all objects (c, z) of \(\int _C\mathcal {D}\), if and only if the 2functor \(\bar{F}:\int _AF^*\mathcal {D}\rightarrow \int _C\mathcal {D}\) has the property \(\mathrm {B}_l\), the equivalence \((b)\Leftrightarrow (c)\) is proven.
The equivalence \((a)\Leftrightarrow (b)\) follows from the fact that, for any 1cell \((h,w):(c,z)\rightarrow (c',z')\) in \(\int _C\mathcal {D}\), the square of 2functors
commutes, where, recall, both vertical 2functors \(\bar{\pi }\) are weak equivalences. So, the 2functors \(\overline{(h,w)}_*\) at the top are weak equivalences if and only if the 2functors \(\bar{h}_*\) at the bottom are as well. This directly means that \((a)\Rightarrow (b)\), and the converse follows from taking any \(\mathcal {D}\) such that \(\mathcal {D}_c\ne \emptyset \) for all \(c\in C\).
Next, we prove that \((c)\Rightarrow (d)\): For any object (c, z) of \(\int _C\mathcal {D}\), we have the squares
whose composite is the left square in (10), and where \(j_z(1_c)=(c,z)\). By hypothesis, the left square is a homotopy pullback. Furthermore, as F has the property \(\mathrm {B}_l\), owing to the already proven implication \((c)\Rightarrow (a)\), Theorem 2.3 implies that the composite square is also a homotopy pullback. Therefore, by the homotopy fibre characterization, the right square above is a homotopy pullback as well.
Finally, \((d)\Rightarrow (a)\) is easy: For any object \(c\in C\) take \(\mathcal {D}=C(,c):C^{\mathrm {op}}\rightarrow {\underline{\mathbf{Cat}}}\subseteq \) \({\underline{\mathbf{2Cat}}}\). Then, by hypothesis, the square
is a homotopy pullback, whence the result follows from Theorem 2.3. \(\square \)
Likewise, if \(F:A\rightarrow C\) and \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) are 2functors, for any objects \(c\in C\) and \(z\in \mathcal {D}_{\!c}\), we have the 2functor
given by
and the result below holds.
Theorem 5.3
For a 2functor \(F:A\rightarrow C\), the following statements are equivalent:

(a)
\(F:A\rightarrow C\) has the property \(\mathrm {B}_r\).

(b)
For any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the 2functor \(\bar{F}:\int _AF^*\mathcal {D}\rightarrow \int _C\mathcal {D}\) has the property \(\mathrm {B}_r\).

(c)
For any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), and any objects \(c\in C\) and \(z\in \mathcal {D}_{\!c}\), the commutative square
is a homotopy pullback.

(d)
For any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the square (30)
is a homotopy pullback.
A main consequence is the corollary below.
Corollary 5.1
(Homotopy Cofinality Theorem) Let \(F:A\rightarrow C\) be a 2functor between 2categories. The statements below are equivalent.

(a)
\(F:A\rightarrow C\) is homotopy left (resp. right) cofinal.

(b)
For any 2functor \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (resp. \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) the induced 2functor \(\bar{F}:\int _AF^*\mathcal {D}\rightarrow \int _C\mathcal {D}\) is homotopy left (resp. right) cofinal.

(c)
For any 2functor \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\) (resp. \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\)) the induced 2functor \(\bar{F}:\int _AF^*\mathcal {D}\rightarrow \int _C\mathcal {D}\) is a weak equivalence.
Proof
The equivalence \((a)\Leftrightarrow (b)\) follows from Theorems 5.2 and 5.3. The implication \((b)\Rightarrow (c)\) follows from Theorem 2.4. To prove the remaining \((c)\Rightarrow (a)\), take, for any object c of \(\mathcal {C}\), the 2functor \(\mathcal {D}=C(,c):C^{\mathrm {op}}\rightarrow {\underline{\mathbf{Cat}}}\subseteq {\underline{\mathbf{2Cat}}}\). Then, by hypothesis, the 2functor
is a weak equivalence. Therefore, \(F\!\downarrow \!c\) is weakly contractible as \(C\!\downarrow \!c\) is, by Lemma 2.1. \(\square \)
Next, we show conditions on a 2category C in order for the square (30) to always be a homotopy pullback.
Corollary 5.2
Let C be a 2category. Then, the following properties are equivalent:

(i)
For any 1cell \(h:c\rightarrow c'\) and any object x of C, the functor \(h_*:C(x,c)\rightarrow C(x,c')\) is a weak equivalence.

(i’)
For any 1cell \(h:c\rightarrow c'\) and any object x of C, the functor \(h^*:C(c',x)\rightarrow C(c,x)\) is a weak equivalence.

(ii)
For any two objects \(c,c'\in C\), the canonical square
is a homotopy pullback.^{Footnote 8}

(ii’)
For any two objects \(c,c'\in C\), the canonical square
is a homotopy pullback.

(iii)
For any 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the 2functor \({h}_*:\mathcal {D}_{\!c}\rightarrow \mathcal {D}_{\!c'}\) induced for any 1cell \(h:c\rightarrow c'\) of C is a weak equivalence.

(iii’)
For any 2functor \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the 2functor \({h}^*:\mathcal {D}_{\!c'}\rightarrow \mathcal {D}_{\!c}\) induced for any 1cell \(h:c\rightarrow c'\) of C is a weak equivalence.

(iv)
For any 2functors \(F:A\rightarrow C\) and \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the square
is a homotopy pullback.

(iv’)
For any 2functors \(F:A\rightarrow C\) and \(\mathcal {D}:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\), the square
is a homotopy pullback.

(v)
Any 2functor \(F:A\rightarrow C\) has the property \(\mathrm {B}_r\).

(v’)
Any 2functor \(F:A\rightarrow C\) has the property \(\mathrm {B}_l\).
Proof
\((\mathrm{i})\Leftrightarrow (\mathrm{ii})\):^{Footnote 9} Let \(c:\star \rightarrow C\) be the 2functor given for any object \(c\in C\). Then, for any object \(x\in C\), there is quite an obvious natural isomorphism
between the homotopyfibre 2category (actually a category) of \(c:\star \rightarrow C\) over x and the homcategory C(c, x). Then, any 1cell \(h:x\rightarrow y\) in C induces a weak equivalence \(h_*:c\!\downarrow \!x\rightarrow c\!\downarrow \!y\) if and only if the induced \(h_*:C(c,x)\rightarrow C(c,y)\) is a weak equivalence. It follows that the 2functor \(c:\star \rightarrow C\) has the property \(\mathrm {B}_l\) if and only if the hypothesis in \((\mathrm{i})\) holds. On the other hand, by Theorem 2.3, the 2functor \(c:\star \rightarrow C\) has the property \(\mathrm {B}_l\) if and only if, for any object \(c'\in C\), the square
is a homotopy pullback, that is, if and only if \((\mathrm{ii})\) holds.
\((\mathrm{iii})\Rightarrow (\mathrm{i})\): For any object \(x\in C\), the result follows by applying the hypothesis in \((\mathrm{iii})\) to the 2functor \(\mathcal {D}=C(x,):C\rightarrow {\underline{\mathbf{Cat}}}\).
\((\mathrm{i})\Rightarrow (\mathrm{v})\) By Theorem 5.1, for any 2functor \(F:A\rightarrow C\) and any object \(c\in C\), the square
is a homotopy pullback. Then, F has the property \(\mathrm {B}_r\) by Theorem 2.3.
\((\mathrm{v})\Rightarrow (\mathrm{iv})\) This follows from Theorem 5.3.
\((\mathrm{iv})\Rightarrow (\mathrm{iii})\) This follows from Theorem 5.1.
Thus, we have \((\mathrm{i})\Leftrightarrow (\mathrm{ii})\Leftrightarrow (\mathrm{iii})\Leftrightarrow (\mathrm{iv})\Leftrightarrow (\mathrm{v})\) and, similarly, we also have the equivalences \((\mathrm{i}')\Leftrightarrow (\mathrm{ii}')\Leftrightarrow (\mathrm{iii}')\Leftrightarrow (\mathrm{iv}')\Leftrightarrow (\mathrm{v}')\).
Furthermore, for any given 2functor \(F:A\rightarrow C\), the application of the hypothesis in \((\mathrm{iii})\) to the homotopyfibre 2functor \(F\!\downarrow \!:C\rightarrow \) \({\underline{\mathbf{2Cat}}}\) just says that F has the property \(\mathrm {B}_l\). Hence, \((\mathrm{iii})\Rightarrow (\mathrm{v}')\). Likewise, for any 2functor \(F:A\rightarrow C\), the hypothesis on \(\!\downarrow \!F:C^{\mathrm {op}}\rightarrow \) \({\underline{\mathbf{2Cat}}}\) implies that F has the property \(\mathrm {B}_r\), whence \((\mathrm{iii}')\Rightarrow (\mathrm{v})\), and the proof is complete. \(\square \)
Remark 1
If \(\phi :A\rightarrow C\) and \(\phi ':A'\rightarrow C\) are continuous maps between spaces, its homotopyfibre product \(A\times ^{_\mathrm {h}}_CA'\) is the subspace of the product \(A\times B^I\!\times A'\), where \(I=[0,1]\) and \(C^{I}\) is taken with the compactopen topology, whose points are triples \((a,\gamma ,a')\) with \(a\in A\), \(a'\in A'\), and \(\gamma : \phi a\rightarrow \phi 'a'\) a path in C joining \(\phi a\) and \(\phi ' a'\), that is, \(\gamma :I\rightarrow C\) is a path with \(\gamma 0=\phi a\) and \(\gamma 1=\phi ' a'\).
There is a subtle 2categorical emulation of the homotopyfibre product of spaces: Any pair of 2functors \(F:A\rightarrow C\) and \(F':A'\rightarrow C\), where A, C, and \(A'\) are 2categories, has associated a comma 2category \(F\!\downarrow \!F'\), whose 0cells are triples \((a,f,a')\) with \(f:Fa\rightarrow F'a'\) a 1cell in C, whose 1cells \((u,\beta ,u'):(a_0,f_0,a'_0)\Rightarrow (a_1,f_1,a'_1)\) are triples consisting of 1cells \(u:a_0\rightarrow a_1\) in A and \(u':a'_0\rightarrow a'_1\) in \(A'\), together with a 2cell \(\beta : F'u'\circ f_0\Rightarrow f_1\circ Fu\) in C, and 2cells \((\alpha ,\alpha '):(u,\beta ,u')\Rightarrow (v,\gamma ,v')\) pairs given by 2cells \(\alpha :u\Rightarrow v\) in A and \(\alpha ':u'\Rightarrow v'\) in \(A'\) such that \((1_{f_1}\circ F\alpha )\cdot \beta =\gamma \cdot (F'\alpha '\circ 1_{f_0})\). This 2category \(F\!\downarrow \!F'\) comes with a canonical map from its classifying space to the homotopyfibre product space of the induced maps \(\!F\!:\!A\!\rightarrow \!C\!\) and \(\!F'\!:\!A'\!\rightarrow \!C \!\),
which, by [15, Theorem 3.8], is a homotopy equivalence whenever the equivalent conditions of Corollary 5.2 on the 2category C hold.
To finish, let us remark that the class of 2categories satisfying the conditions in Corollary 5.2 above includes those 2categories C where, for each 1cell \(f:c\rightarrow c'\), there exists a 1cell \(f':c'\rightarrow c\) such that \([f'\circ f]=[1_c]\in \pi _0C(c,c)\) and \([f\circ f']=[1_{c'}]\in \pi _0C(c',c')\). In particular, the result applies to 2groupoids, whose 1cells are all invertible, which, recall, are equivalent to crossed modules over groupoids by Brown and Higgins [8].
Notes
There is another model structure on \(\mathbf {2}\mathbf {Cat}\), known as the naive, or folk model structure [27], where weak equivalences are the biequivalences.
For F a functor between small categories, this condition is referred by Cisinski in [19, 6.4.1] by saying that “the functor F is locally constant”.
For F a functor between small categories, this condition is referred by Cisinski in [19, 3.3.3] by saying that “F is aspherical”.
For a 2functor \(\mathcal {D}:C\rightarrow \) \({\underline{\mathbf{Cat}}}\) \(\subseteq \) \({\underline{\mathbf{2Cat}}}\), the simplicial category \(\mathcal {B}_C\mathcal {D}\) is called the homotopy colimit of D, and denoted \(\mathrm {hocolim}_C\mathcal {D}\), by Hinich and Schechtman in [25, Definition (2.2.2)].
Warning: \(\mathcal {B}_C\mathcal {D}\ne \mathcal {B}_{C^{\mathrm {op}}}\mathcal {D}\).
This implies that, for any object \(c\in C\), \(C(c,c)=\Omega (C,c)\); that is, the category C(c, c) is a loop object for the pointed 2category (C, c).
The implication \((\mathrm{i})\Rightarrow (\mathrm{ii})\) was proven by Del Hoyo in [20, Theorem 8.5].
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Acknowledgments
The authors are much indebted to the referee, whose useful observations greatly improved our exposition. This work has been supported by DGI of Spain, Project MTM201122554. Also, the second author by FPU Grant AP20103521 and partly by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the Project UID/MAT/00297/2013 (Centro de Matemática e Aplicações).
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Communicated by Tim Porter and George Janelidze.
Dedicated to Ronald Brown on his 80th birthday.
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Cegarra, A.M., Heredia, B.A. Homotopy colimits of 2functors. J. Homotopy Relat. Struct. 11, 735–774 (2016). https://doi.org/10.1007/s4006201601502
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DOI: https://doi.org/10.1007/s4006201601502