Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories

Abstract

We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield–Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.

If ${\mathscr {C}}$ is a model category (we assume all model categories to be complete and cocomplete) and $\varGamma$ a (small) category, we denote by ${\mathscr {C}}^{\varGamma } = \textrm{Fun}( \varGamma , {\mathscr {C}})$ the category of functors $\varGamma \rightarrow {\mathscr {C}}$, which we shall also refer to as “diagrams” of shape $\varGamma$. It is natural to call a map of diagrams $\alpha :F \rightarrow G$ in ${\mathscr {C}}^{\varGamma }$ a weak equivalence if $\smash {\alpha _{\gamma }} :F(\gamma ) \rightarrow G(\gamma )$ is a weak equivalence in ${\mathscr {C}}$ for all objects $\gamma \in \varGamma$. We shall refer to such weak equivalences as componentwise weak equivalences. But then we immediately run into the problem that the limit functor  does not in general preserve weak equivalences. Since  is a right adjoint, this leads us into trying to derive it. The right derived functor of  is called the homotopy limit and is denoted . Dually, the left derived functor of  is called the homotopy colimit and is denoted .

For many purposes, the abstract existence of homotopy limits is all you need. However, there are also many cases where a concrete, minimalistic realization of them is useful for working with abstract notions. For instance, this paper grew out of an attempt to concretize a concept from derived algebraic geometry. More specifically, we wanted to develop a homological algebra model for the dg-category of quasi-coherent sheaves on a dg-scheme which are equivariant with respect to the action of a group dg-scheme. This question was addressed in Block et al. [2] where a partial result was obtained under serious restrictions (Proposition 13). The general case was stated as a conjecture, see Conjecture 1 in the same paper. In the companion paper to this one, Arkhipov and Ørsted [1], we cover the general case and prove that conjecture (see Theorem 4.1.1), and the key result of homotopical nature is proved in the present paper (see Example 5.4).

Quillen’s model category machinery tells us how to derive the limit: We must equip the diagram category ${\mathscr {C}}^{\varGamma }$ with a model structure with componentwise weak equivalences and in which the limit functor  is a right Quillen functor. In this case, the derived functor is given by for some fibrant replacement $R(F)$ in $\smash {{\mathscr {C}}^{\varGamma }}$. Indeed, such a model structure on ${\mathscr {C}}^{\varGamma }$ exists e.g. if the model category ${\mathscr {C}}$ is combinatorial (see Lurie [7], Propositions A.2.8.2 and A.2.8.7). More precisely, we have the injective model structure  $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Inj}} }$ where weak equivalences and cofibrations are calculated componentwise.

The injective model structure being in general rather complicated, calculating such a replacement of a diagram in practice becomes very involved for all but the simplest shapes of the category $\varGamma$. Therefore, traditionally, other tools have been used. One of the most popular techniques involves adding a parameter to the limit functor  before deriving it. The result is the end bifunctor $\int _{\varGamma } :\varGamma ^{\textrm{op}}\times \varGamma \rightarrow {\mathscr {C}}$ (introduced below) which is in general much easier to derive.

One of the classical accounts of this technique is Hirschhorn [5] who mainly works in the setting of simplicial model categories, which are model categories enriched over simplicial sets, and which furthermore are equipped with a powering functor

$$\begin{aligned} \textrm{SSet}^{\textrm{op}}\times {\mathscr {C}}\longrightarrow {\mathscr {C}}, \qquad ( K, c ) \longmapsto c^{K}, \end{aligned}$$

and a copowering functor

$$\begin{aligned} \textrm{SSet}\times {\mathscr {C}}\longrightarrow {\mathscr {C}}, \qquad ( K, c ) \longmapsto K\otimes c, \end{aligned}$$

satisfying some compatibility relations with the model structure. He then establishes the classical Bousfield–Kan formula

(0.1)

where we write $N( \varGamma / \gamma )$ for the nerve of the comma category of maps in $\varGamma$ with codomain $\gamma$. Or rather, he uses this formula as his definition of homotopy limits (see definition 18.1.8). A proof that this formula agrees with the general definition of homotopy limits is due to Gambino [4, equation (2)] using the machinery of Quillen 2-functors.

Hirschhorn [5] then generalizes this formula to arbitrary model categories in chapter 19, definition 19.1.5. He shows that even for non-simplicial model categories, one can replace simplicial powerings and copowerings by a weaker notion, unique up to homotopy in a certain sense. He then takes the formula (0.1) to be his definition of a homotopy limit.

This paper is devoted to proving (see Theorem 4.2) that indeed, the formula (0.1) agrees with the general definition of a homotopy limit (similarly to what Gambino [4] did in the simplicial setting), at least in the case when the model category ${\mathscr {C}}$ is combinatorial. (This assumption is not a great problem since combinatoriality is what provides us with a model structure on the category ${\mathscr {C}}^{\varGamma }$ in the first place, which is what guarantees that homotopy limits as we defined them above make sense). To the authors’ knowledge, such a proof has not been carried out in the literature before. Finally, in Sect. 5, we discuss preservation of homotopy limits by homotopy-initial functors, observing why this classical property can also be proved using our machinery. As an application, we obtain the classical and fundamental fat totalization formula in the previously mentioned Example 5.4.

1 The Projective and Injective Model Structures

If ${\mathscr {C}}$ is a model category and $\varGamma$ any category, it is natural to demand of any model structure on the functor category ${\mathscr {C}}^{\varGamma } = \textrm{Fun}( \varGamma , {\mathscr {C}})$ that weak equivalences must be calculated componentwise. The two most natural model structures one can hope for (which may or may not exist) are

• The projective model structure $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} }$ where weak equivalences and fibrations are calculated componentwise.

• The injective model structure $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Inj}} }$ where weak equivalences and cofibrations are calculated componentwise.

Both model structures are known to exist when ${\mathscr {C}}$ is a combinatorial model category, see Lurie [7, Proposition A.2.8.2]. We shall also use the attributes “projective(ly)” and “injective(ly)” when referring to these model structures, so e.g. “projectively cofibrant” means cofibrant in the projective model structure.

Proposition 1.1

(Lurie [7, Proposition A.2.8.7]) If ${\mathscr {C}}$ is a model category and $f :\varGamma \rightarrow \varGamma '$ a functor, denote by $f^{*}$ the restriction functor $\smash { {\mathscr {C}}^{\varGamma '} \rightarrow {\mathscr {C}}^{\varGamma } }$. Then $f^{*}$ fits as the right and left adjoint of Quillen adjunctions

whenever the model structures in question exist. In particular, we see that the adjunctions and are Quillen adjunctions.

The adjoints $f_{!}$ and $f_{*}$ are the usual left and right Kan extensions along $f$, which are given by limits

(1.2)

These limits are taken over the categories of maps $f(\gamma )\rightarrow \gamma '$ (resp. $\gamma '\rightarrow f(\gamma )$) in $\varGamma '$ for varying $\gamma \in \varGamma$.

Corollary 1.3

Assume that the projective model structure $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} }$ exists.

• (i) If $\varphi :c \rightarrow c'$ is a (trivial) cofibration in ${\mathscr {C}}$ and $\gamma _{0} \in \varGamma$ is an object, then the coproduct map$\coprod _{\!\!\!\varGamma ( \gamma _{0},\mathord {-}) } \varphi :\!\!\!\!\!\coprod _{\!\!\!\varGamma ( \gamma _{0},\mathord {-}) } c \rightarrow \!\!\!\!\!\coprod _{\!\!\!\varGamma ( \gamma _{0},\mathord {-}) } c'$ is a (trivial) cofibration in $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} }$. We shall refer to such (trivial) cofibrations as simple (trivial) projective cofibrations.

• (ii) If $f :\varGamma \rightarrow \varGamma '$ is a functor, then $f_{!} :{\mathscr {C}}^{\varGamma }_{\textrm{Proj}} \rightarrow {\mathscr {C}}^{\varGamma '}_{\textrm{Proj}}$ preserves simple (trivial) projective cofibrations, taking ${\!\!\!\coprod }_{\!\!\!\varGamma ( \gamma _{0},\mathord {-}) } \varphi$ to ${\!\!\!\coprod }_{\!\!\!\varGamma '( f( \gamma _{0} ),\mathord {-}) } \varphi$.

There is, of course, a completely dual statement for (trivial) fibrations, with $\coprod _{\!\!\!\varGamma (\gamma _0,\mathord {-})}$ replaced by $\prod _{\varGamma (\mathord {-},\gamma _0)}$, and we shall dually use the term simple (trivial) fibrations. We may note that what we call the simple (trivial) cofibrations form a generating set for the model category $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} }$, whereas the situation is more complicated for $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Inj}} }$, see Lurie [7, Proposition A.2.8.2].

Proof

Applying Proposition 1.1 to the embedding $\iota :\gamma _{0} \hookrightarrow \varGamma$ of the full subcategory with $\gamma _{0}$ as the only object, we get that $\iota _{!} \varphi$ is a (trivial) cofibration. Now $\smash { \iota _{!} \varphi = {\!\!\!\coprod }_{\!\!\!\varGamma ( \gamma _{0}, \mathord {-}) } \varphi }$ by the above colimit formula for left Kan extension. The statement (ii) follows by applying Kan extensions to the diagram

and using that Kan extensions, being adjoints to restriction, respect compositions. $\square$

2 The Reedy Model Structure

A third approach exists to equipping diagram categories ${\mathscr {C}}^{\varGamma }$ with a model structure, provided the category $\varGamma$ has the structure of a Reedy category. A category $\varGamma$ is called Reedy if it contains two subcategories $\varGamma _{+}, \varGamma _{-} \subset \varGamma$, each containing all objects, such that

• there exists a degree function ${{\,\textrm{Ob}\,}}\varGamma \rightarrow {\mathbb {Z}}$, such that non-identity morphisms from $\varGamma _{+}$ strictly raise the degree and non-identity morphisms from $\varGamma _{-}$ strictly lower the degree (more generally, an ordinal number can be used instead of ${\mathbb {Z}}$);

• each morphism $f \in \varGamma$ factors uniquely as $f = g h$ for $g \in \varGamma _{+}$ and $h \in \varGamma _{-}$.

We note that a direct category is Reedy with $\varGamma _{+} = \varGamma$, and that an inverse category is Reedy with $\varGamma _{-} = \varGamma$. If $\varGamma$ is Reedy, then so is $\varGamma ^{\textrm{op}}$, with $( \varGamma ^{\textrm{op}} )_{+} = ( \varGamma _{-} )^{\textrm{op}}$ and $( \varGamma ^{\textrm{op}} )_{-} = ( \varGamma _{+} )^{\textrm{op}}$.

Example 2.1

The simplex category ${\varvec{\mathrm {\Delta }}}$ is Reedy with ${\varvec{\mathrm {\Delta }}}_{+}$ consisting of injective maps and ${\varvec{\mathrm {\Delta }}}_{-}$ consisting of surjective maps. The degree function does the obvious thing, $[n] \mapsto n$.

If $\varGamma$ is a Reedy category and ${\mathscr {C}}$ is any model category, and if $F \in {\mathscr {C}}^{\varGamma }$ is a diagram, we define the latching and matching objects by

In other words, the limit (resp. colimit) runs over the category of all non-identity maps $\alpha \rightarrow \gamma$ in $\varGamma _{+}$ (resp. $\gamma \rightarrow \alpha$ in $\varGamma _{-}$). The latching map is the canonical map $\smash { L_{\gamma } F \rightarrow F(\gamma ) }$, and the matching map is the canonical map $\smash { F(\gamma ) \rightarrow M_{\gamma } F }$.

If $f :F \rightarrow G$ is a map in ${\mathscr {C}}^{\varGamma }$, then the relative latching map and the relative matching map are the maps

$$\begin{aligned} F(\gamma ) \mathbin {\textstyle \coprod \limits _{L_{\gamma } F}} L_{\gamma } G \longrightarrow G(\gamma ) \qquad \text {resp.}\qquad F(\gamma ) \longrightarrow G(\gamma ) \mathbin {\textstyle \prod \limits _{M_{\gamma } G}} M_{\gamma } F \end{aligned}$$

given by the universal property of the pushout resp. pullback. We say that $f$ is a (trivial) Reedy cofibration (resp. fibration) if the relative latching (resp. matching) map is a (trivial) cofibration (resp. fibration) in ${\mathscr {C}}$. If $F = \varnothing$ (resp. $F = *$), we recover the latching (resp. matching) map. For our arbitrary model category ${\mathscr {C}}$, this defines a model structure on $\smash {{\mathscr {C}}^{\varGamma }}$, called the Reedy model structure, see Hirschhorn [5, Theorem 15.3.4]. The weak equivalences are componentwise weak equivalences. We shall write $\smash {{\mathscr {C}}^{\varGamma }_{\textrm{Reedy}}}$ when we equip the diagram category with this model structure.

3 Homotopy Limits

We recall the following definition, referring to e.g. Riehl [8] for further details. Let $\varGamma$ and ${\mathscr {C}}$ be categories with ${\mathscr {C}}$ complete and cocomplete, and let $H :\varGamma ^{\textrm{op}} \times \varGamma \rightarrow {\mathscr {C}}$ be a bifunctor. The end of $H$ is an object $\smash { \int _{\varGamma } H = \int _{\gamma \in \varGamma } H(\gamma ,\gamma ) }$ in ${\mathscr {C}}$, together with morphisms $\int _{\varGamma } H \rightarrow H(\gamma ,\gamma )$ for all $\gamma \in \varGamma$, such that for any $f :\gamma \rightarrow \gamma '$, the following diagram commutes:

Furthermore, $\int _{\varGamma } H$ is universal with this property in the sense that if $A$ is another object of ${\mathscr {C}}$ with a collection of arrows $A \rightarrow H(\gamma ,\gamma )$ for all $\gamma$, subject to the same commutativity conditions, then these factor through a unique arrow $\smash {A\rightarrow \int _{\varGamma } H}$. There is a dual notion of a coend, denoted instead by $\smash { \int ^{\varGamma } H }$, which we shall not spell out.

Remark 3.1

A diagram $F \in {\mathscr {C}}^{\varGamma }$ may be regarded as a diagram in ${\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma }$ which is constant with respect to the first variable. In that case, it follows from the universal property of the end that recovers the limit of the diagram.

Remark 3.2

The end fits as the right adjoint of the adjunction

The left adjoint takes $A \in {\mathscr {C}}$ to the bifunctor ${\!\!\!\coprod }_{\!\!\!{{\,\textrm{Hom}\,}}_{\varGamma } ( \mathord {-}, \mathord {-}) } A :\varGamma ^{\textrm{op}}\times \varGamma \rightarrow {\mathscr {C}}$. In the literature, this is usually written as an adjunction

$$\begin{aligned} \textrm{Set}^{\varGamma ^{\textrm{op}}\times \varGamma } \bigl ( {{\,\textrm{Hom}\,}}_{\varGamma }, {\mathscr {C}}(A,F) \bigr ) \cong {\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma } \bigl ( A, \textstyle \int _{\varGamma } F \bigr ) \end{aligned}$$

for $A \in {\mathscr {C}}$, which is just the well-known statemant that the end is the weighted limit ${\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma } \rightarrow {\mathscr {C}}$ with weight ${{\,\textrm{Hom}\,}}_{\varGamma }$.

The following theorem is the basis for all our homotopy limit formulae:

Theorem 3.3

Let ${\mathscr {C}}$ be a model category and $\varGamma$ a category. Regard the functor category $\smash { {\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma } }$ as a model category in any of the following ways:

• (i) as ${\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma } = ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Proj}} )^{\varGamma }_{\textrm{Inj}}$ (assuming this model structure exists);

• (ii) as ${\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma } = ( {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} )^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}}$ (assuming this model structure exists);

• (iii) as ${\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma } = {\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma }_{\textrm{Reedy}}$ (assuming $\varGamma$ is Reedy).

Then the end functor $\int _{\varGamma } :{\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma } \rightarrow {\mathscr {C}}$ is right Quillen.

Proof

We initially prove the first statement, the second one being dual. By Remark 3.2, it suffices to check that the left adjoint $\coprod _{{\text {\!\!\!Hom}}_{\varGamma }}$ takes (trivial) cofibations in ${\mathscr {C}}$ to (trivial) cofibations in $\smash { ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Proj}} )^{\varGamma }_{\textrm{Inj}} }$. If $c \rightarrow c'$ is a (trivial) cofibration in ${\mathscr {C}}$, then we must therefore consider the map $\mathop {\!\!\!\coprod }_{\!\!\!\varGamma (- ,- ) } c \rightarrow \mathop {\!\!\!\coprod }_{\!\!\!\varGamma (- ,- ) } c'$ in $\smash { ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Proj}} )^{\varGamma }_{\textrm{Inj}} }$. Checking that this is a (trivial) injective cofibration over $\varGamma$ amounts, by definition, to checking this componentwise. But for a fixed $\gamma _{0} \in \varGamma$, this component is $\mathop {\!\!\!\coprod }_{\!\!\!\varGamma ( \_ , \gamma _{0} ) } c \rightarrow \mathop {\!\!\!\!\!\!\coprod }_{\!\!\!\varGamma (-,\gamma _{0})} c'$ which is a simple (trivial) projective cofibration in $\smash { {\mathscr {C}}^{\varGamma ^{\textrm{op}}} }$.

For the Reedy case, we recall from Lurie [7, Example A.2.9.22] and Hirschhorn [5, Theorem 15.5.2] that being a (trivial) cofibration in the model category $\smash { {\mathscr {C}}^{\varGamma \times \varGamma ^{\textrm{op}}}_{\textrm{Reedy}} } = \smash { ( {\mathscr {C}}^{\varGamma }_{\textrm{Reedy}} ) ^ {\varGamma ^{\textrm{op}}} _ {\textrm{Reedy}} }$ is equivalent to the restriction being a (trivial) cofibration in

$$\begin{aligned} \smash { \bigl ( {\mathscr {C}}^{\varGamma _{+}}_{\textrm{Proj}} \bigr ) }^{(\varGamma ^{\textrm{op}})_{+}}_{\textrm{Proj}} = \smash { \bigl ( {\mathscr {C}}^{\varGamma _{+}}_{\textrm{Proj}} \bigr ) }^{ ( \varGamma _{-} )^{\textrm{op}} }_{\textrm{Proj}}. \end{aligned}$$

But we have, by the unique factorization property of Reedy categories, that

$$\begin{aligned} \!\!\coprod _{ \varGamma (\mathord {-},\mathord {-}) } c =\!\!\coprod _{ \gamma _{0} \in \varGamma } \!\!\coprod _{ \varGamma _{-} ( \mathord {-}, \gamma _{0} ) } \coprod _{ \varGamma _{+} ( \gamma _{0},\mathord {-}) } c \end{aligned}$$

for any $c\in {\mathscr {C}}$. These consist of coproducts of exactly the same form as the ones appearing in the definition of simple (trivial) projective cofibrations (Corollary 1.3 (i)). Thus we find that for any (trivial) cofibration $c\rightarrow c'$ in ${\mathscr {C}}$, the map ${\!\!\coprod }_{\!\!\!\varGamma (\mathord {-},\mathord {-}) } c \rightarrow {\!\!\coprod }_{\!\!\!\varGamma (\mathord {-},\mathord {-}) } c'$ is a (trivial) cofibration in $\smash { {\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma }_{\textrm{Reedy}} }$. $\square$

Thus we can derive the end using any of these three model structures, when available. Write ${\mathbb {R}}\!\!\int _{\varGamma } :{\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma }\rightarrow {\mathscr {C}}$ for the derived functor, which we shall call the homotopy end.

Corollary 3.4

If ${\mathscr {C}}$ is a combinatorial model category and $\varGamma$ a category, then for a diagram $F\in {\mathscr {C}}^{\varGamma }$,

where $R$ is a fibrant replacement with respect to the model structure $\smash { ( {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} )^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}} }$ or, if $\varGamma$ is Reedy, in $\smash { {\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma }_{\textrm{Reedy}} }$.

Proof

First write for some fibrant replacement functor $R_{\varGamma }$ in $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Inj}} }$. Now Proposition 1.1 and Lurie [7, Remark A.2.8.6] show that the constant functor embedding $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Inj}} \hookrightarrow ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Proj}})^{\varGamma }_{\textrm{Inj}} }$ is right Quillen and thus preserves fibrant objects. Thus $R_{\varGamma }(F)$ is also fibrant in $\smash { ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Proj}} )^{\varGamma }_{\textrm{Inj}} }$. This proves the first equality sign. The second one is clear. $\square$

Of course, even though $F$ as a diagram in ${\mathscr {C}}^{\varGamma ^{\textrm{op}}\times \varGamma }$ was constant with respect to the first variable, $R(F)$ is in general not. Remarkably, since ends calculate naturality between the two variables, this often makes calculations of homotopy limits more manageable, compared to resolving the diagram inside $\smash { {\mathscr {C}}^{\varGamma }_{\textrm{Inj}} }$.

Corollary 3.5

Suppose $\varGamma$ is a direct category, and let $\smash { R :{\mathscr {C}}\rightarrow {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}} }$ be a functor that takes $c\in {\mathscr {C}}$ to a fibrant replacement of the constant diagram at $c$. Then

Proof

Clearly, $R(F)$ is a fibrant replacement inside $\smash { ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}} ) ^ {\varGamma }_{\textrm{Proj}} }$. By Lurie [7, Example A.2.9.22] and Hirschhorn [5, Theorem 15.5.2], this model category is equal to

$$\begin{aligned} \smash { \bigl ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}} \bigr ) } ^{\varGamma }_{\textrm{Proj}} = \smash { \bigl ( {\mathscr {C}}^{\varGamma ^{\textrm{op}}}_{\textrm{Reedy}} \bigr ) }^{\varGamma }_{\textrm{Reedy}} = \smash { \bigl ( {\mathscr {C}}^{\varGamma }_{\textrm{Reedy}} \bigr ) }^{\varGamma ^{\textrm{op}}}_{\textrm{Reedy}} = \smash { \bigl ( {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} \bigr ) }^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}}, \end{aligned}$$

so the result follows from Theorem 3.3. $\square$

4 Bousfield–Kan Formula

In Hirschhorn [5, chapter 19], homotopy limits are being developed for arbitrary model categories via a machinery of simplicial resolutions. In this section, we use Theorem 3.3/Corollary 3.4 to explain why this machinery works. Throughout, we denote by $\textrm{SSet}$ the category of simplicial sets endowed with the Quillen model structure.

If ${\mathscr {C}}$ is a (complete) category and  a simplicial diagram in ${\mathscr {C}}$, we may extend  to a continuous functor $X :\textrm{SSet}^{\textrm{op}} \rightarrow {\mathscr {C}}$ via the right Kan extension along the Yoneda embedding ${\varvec{\mathrm {\Delta }}}^{\textrm{op}} \hookrightarrow \textrm{SSet}^{\textrm{op}}$:

If ${\mathscr {C}}$ is a model category, the matching object at $[n]$ is  , and so  being Reedy-fibrant is equivalent to the map $\smash { X_{n} = X^{\varDelta ^{\!n}} \rightarrow X^{\partial \varDelta ^{\!n}} }$ being a fibration in ${\mathscr {C}}$ for all $n$.

We need the following technical lemma:

Lemma 4.1

(Hovey [6, Proposition 3.6.8]) Let ${\mathscr {C}}$ be a model category and $F :\textrm{SSet}\rightarrow {\mathscr {C}}$ a functor preserving colimits and cofibrations. Then $F$ preserves trivial cofibrations if and only if $\smash { F( \varDelta ^{\!n} ) \rightarrow F( \varDelta ^{\!0} ) }$ is a weak equivalence for all $n$.

Theorem 4.2

(Bousfield–Kan formula) Suppose ${\mathscr {C}}$ is a combinatorial model category, $\varGamma$ a category, and $F \in {\mathscr {C}}^{\varGamma }$. Let $\smash { R :{\mathscr {C}}\rightarrow {\mathscr {C}}^{ {\varvec{\mathrm {\Delta }}}^{\textrm{op}} } }$ be a functor that takes $c \in {\mathscr {C}}$ to a Reedy-fibrant replacement of the constant ${\varvec{\mathrm {\Delta }}}^{\textrm{op}}$-diagram at $c$. Let furthermore $\smash { K \in \textrm{SSet}^{\varGamma }_{\textrm{Proj}} }$ be a projectively cofibrant resolution of the point. Then

Proof

Clearly,  is a fibrant replacement of $F$ with respect to the model structure $\smash { ( {\mathscr {C}}^{{\varvec{\mathrm {\Delta }}}^{\textrm{op}}}_{\textrm{Reedy}} ) ^{\varGamma }_{\textrm{Proj}} }$. The theorem will follow if we prove that $R(F(\mathord {-}))^{K(\mathord {-})}$ is a fibrant replacements of $F$ in $\smash { ( {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} )^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}} }$. This will follow from Ken Brown’s Lemma if we prove that the continuous functor

$$\begin{aligned} \bigl ( \textrm{SSet}^{\varGamma }_{\textrm{Proj}} \bigr )^{\textrm{op}} \longrightarrow \smash { \bigl ( {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} \bigr ) }^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}}, \qquad K(\mathord {-}) \longmapsto R(F(\mathord {-}))^{K(\mathord {-})}, \end{aligned}$$

takes opposites of (trivial) cofibrations to (trivial) fibrations. (Trivial) cofibrations in $\smash { \textrm{SSet}^{\varGamma }_{\textrm{Proj}} }$ are generated from simple (trivial) projective cofibrations via pushouts and retracts, c.f. Lurie [7, Proposition A.2.8.2]. Thus by continuity of the functor, it suffices to prove the statement for simple (trivial) cofibrations. We therefore let ${\!\!\coprod }_{\!\!\!\varGamma (\gamma _{0},\mathord {-}) } K \hookrightarrow {\!\!\!\coprod }_{\!\!\!\varGamma (\gamma _{0},\mathord {-}) } L$ be one such, where $K \hookrightarrow L$ is a (trivial) cofibration and $\gamma _{0} \in \varGamma$. This is mapped to

$$\begin{aligned}\textstyle \!\!\!\!\!\prod _{ \varGamma (\gamma _{0}, \mathord {-}) } R(F(\mathord {-}))^{L} \rightarrow \!\prod _{ \varGamma (\gamma _{0}, \mathord {-}) } R(F(\mathord {-}))^{K}. \end{aligned}$$

Thus we must show that the composition

$$\begin{aligned} \textrm{SSet}^{\textrm{op}} \xrightarrow {\prod _{\varGamma (\gamma _{0},\mathord {-})}} \bigl ( \textrm{SSet}^{\varGamma ^{\textrm{op}}}_{\textrm{Proj}} \bigr )^{\textrm{op}} \longrightarrow \smash { \bigl ( {\mathscr {C}}^{\varGamma }_{\textrm{Proj}} \bigr ) }^{\varGamma ^{\textrm{op}}}_{\textrm{Inj}}, \qquad \textstyle K \longmapsto \prod _{ \varGamma (\gamma _{0},\mathord {-}) } R(F(\mathord {-}))^{K}, \end{aligned}$$

takes (trivial) cofibrations to (trivial) fibrations. Checking that it takes cofibrations to fibrations amounts to checking this for the generating cofibrations $\partial \varDelta ^{\!n}\hookrightarrow \varDelta ^{\!n}$ in $\textrm{SSet}$. This holds by the assumption that  is componentwise Reedy-fibrant. Since the functor takes colimits to limits, the claim now follows from the (dual of) the lemma. $\square$

Remark 4.3

One may prove Hirschhorn (see e.g. [5, Proposition 14.8.9]) that the diagram $\smash { K(\mathord {-}) = N( \varGamma /{\mathord {-}} ) \in \textrm{SSet}^{\varGamma }_{\textrm{Proj}} }$, taking $\gamma$ to the nerve $N(\varGamma /\gamma )$ of the comma category $\varGamma /\gamma$ of all maps in $\varGamma$ with codomain $\gamma$, is a projectively cofibrant resolution of the point. Thus we have

(4.4)

which is the classical form of the Bousfield–Kan formula (0.1).

5 Homotopy-Initial Functors

A functor $f :\varGamma \rightarrow \varGamma '$ is called homotopy-initial if for all objects $\gamma ' \in \varGamma '$, the nerve $N(f/\gamma ')$ is contractible as a simplicial set; here $f/\gamma '$ denotes the comma category whose objects are pairs $(\gamma ,\alpha )$ where $\alpha$ is a map $f(\gamma ) \rightarrow \gamma '$. A morphism $( \gamma _1, \alpha ) \rightarrow ( \gamma _2, \alpha )$ is a morphism $\gamma _1 \rightarrow \gamma _2$ in $\varGamma$ making the diagram

commute. We aim to reprove, using our more modern language, the statement that homotopy-initial functors preserve homotopy limits. This relies on a few technical lemmas:

Lemma 5.1

If $f :\varGamma \rightarrow \varGamma '$ is a functor, then $f_{!} N( \varGamma / \mathord {-}) = N(f/\mathord {-}) \in \textrm{SSet}^{\varGamma '}$. In particular, since $\smash { N(\varGamma /\mathord {-}) \in \textrm{SSet}^{\varGamma }_{\textrm{Proj}} }$ is cofibrant, $\smash { N(f/\mathord {-}) \in \textrm{SSet}^{\varGamma '}_{\textrm{Proj}} }$ is cofibrant by Proposition 1.1.

Proof

Since colimits in diagram categories over cocomplete categories can be checked componentwise, this boils down to the observation

$\square$

The following lemma is inspired by Hirschhorn [5, Proposition 19.6.6]. See also Riehl [8, Lemma 8.1.4].

Lemma 5.2

Suppose that ${\mathscr {C}}$ is a complete category and that $\varGamma$ and $\varGamma '$ are two categories with a functor $f :\varGamma \rightarrow \varGamma '$. Then we have

$$\begin{aligned} \int _{\gamma \in \varGamma } F(f(\gamma ))^{N(\varGamma /\gamma )} = \int _{\gamma '\in \varGamma '} F(\gamma ')^{N(f/\gamma ')} \end{aligned}$$

for $F \in ( {\mathscr {C}}^{{\varvec{\mathrm {\Delta }}}^{\textrm{op}}} )^{\varGamma }$ (see the previous chapter for an explanation of the power notation).

Proof

For the purpose of the proof, we recall that the Kan extension formulae in (1.2) may be equivalently written in terms of (co)ends:

$$\begin{aligned} f_{!} F(\gamma ') = \int ^{\gamma \in \varGamma } \varGamma '( f(\gamma ), \gamma ' ) \times F(\gamma ) \qquad \text {and}\qquad f_{*} F(\gamma ') = \int _{\gamma \in \varGamma } F(\gamma )^{\varGamma '(\gamma ',f(\gamma ))}. \end{aligned}$$

Here we are using the natural copowering and powering of $\textrm{Set}$ on ${\mathscr {C}}$, given by $S \times c = \!\!\!\coprod _{\!\!\!S} c$ and $c^{S} = \prod _{S} c$ for $S \in \textrm{Set}$ and $c \in {\mathscr {C}}$, which make sense whenever ${\mathscr {C}}$ is complete resp. cocomplete. We shall furthermore make use of the so-called “co-Yoneda lemma” which says that

$$\begin{aligned} G(f(\gamma )) = \int _{\gamma '\in \varGamma '} G(\gamma ')^{ \varGamma '(f(\gamma ),\gamma ' ) } \qquad \text {for all } G \in {\mathscr {C}}^{\varGamma '}. \end{aligned}$$

Finally, we use “Fubini’s theorem” for ends, which says that ends, being limits, commute. This together yields

$$\begin{aligned} \int _{\gamma \in \varGamma } F(f(\gamma ))^{ N(\varGamma / \gamma ) }&= \int _{\gamma \in \varGamma } \int _{[n]\in {\varvec{\mathrm {\Delta }}}} F(f(\gamma ))_{n}^{ N(\varGamma /\gamma )_{n} } \\&= \int _{\gamma \in \varGamma } \int _{[n]\in {\varvec{\mathrm {\Delta }}}} \int _{\gamma '\in \varGamma '} \bigl ( F(\gamma ')_{n} ^{ \varGamma '( f(\gamma ), \gamma ' ) } \bigr ) ^{ N(\varGamma /\gamma )_{n} } \\&= \int _{\gamma \in \varGamma } \int _{[n]\in {\varvec{\mathrm {\Delta }}}} \int _{\gamma '\in \varGamma '} F(\gamma ')_{n} ^{ \varGamma '( f(\gamma ) , \gamma ' ) \times N(\varGamma /\gamma )_{n} } \\&= \int _{[n]\in {\varvec{\mathrm {\Delta }}}} \int _{\gamma '\in \varGamma '} F(\gamma ')_{n} ^{ \int ^{\gamma \in \varGamma } \varGamma '( f(\gamma ) , \gamma ' ) \times N(\varGamma /\gamma )_{n} } \\&= \int _{\gamma '\in \varGamma '} F(\gamma ')^{ f_{!} N (\varGamma /\mathord {-}) ( \gamma ' ) } = \int _{\gamma '\in \varGamma '} F(\gamma ')^{ N(f/\gamma ') } \end{aligned}$$

where the last equality sign is due to Lemma 5.1. $\square$

Theorem 5.3

(Hirschhorn [5, Theorem 19.6.7]) Suppose ${\mathscr {C}}$ is a combinatorial model category and $\varGamma , \varGamma '$ two categories. If $f :\varGamma \rightarrow \varGamma '$ is homotopy-initial, then we have

for all $F \in {\mathscr {C}}^{\varGamma '}$.

Proof

Theorem 4.2 and Eq. (4.4) show that

Since $N(f/\gamma ')$ is contractible for all $\gamma '$, $N( f / \mathord {-})$ is a projectively cofibrant resolution of the point by Lemma 5.1. Thus the right-hand side is exactly  by Theorem 4.2. $\square$

Example 5.4

(Fat totalization formula) Recall from Example 2.1 that the simplex category ${\varvec{\mathrm {\Delta }}}$ is Reedy with ${\varvec{\mathrm {\Delta }}}_{+}$ being the subcategory containing only injective maps. The inclusion $\smash { \iota :{\varvec{\mathrm {\Delta }}}_{+}\hookrightarrow {\varvec{\mathrm {\Delta }}}}$ is homotopy-initial (see e.g., [8, Example 8.5.12] or [3, Example 21.2]), hence for all . As ${\varvec{\mathrm {\Delta }}}_{+}$ is a direct category, we obtain from Corollary 3.5 that we may calculate  as

for some functor $\smash { R :{\mathscr {C}}\rightarrow {\mathscr {C}}^{{\varvec{\mathrm {\Delta }}}_{+}^{\textrm{op}}} }$ that takes $x$ to an injectively (i.e. Reedy-) fibrant replacement of the constant diagram at $x$ (alternatively, this follows from Theorem 4.2 using the well-known fact that the standard simplex  is projectively cofibrant over ${\varvec{\mathrm {\Delta }}}_{+}$, see e.g. ([8, Example 11.5.6]). This is the so-called fat totalization formula for homotopy limits over ${\varvec{\mathrm {\Delta }}}$. The dual formula for homotopy colimits over ${\varvec{\mathrm {\Delta }}}^{\textrm{op}}$ is called the fat geometric realization formula.

The fat totalization formula is one of the main technical tools used in the companion paper to this one, Arkhipov and Ørsted [1], to develop a homological model for the dg-derived categories of quasi-coherent sheaves on a dg-scheme in terms of the dg-derived categories of quasi-coherent sheaves on a covering. This solves the classical problem that “triangulated categories don’t glue well” entirely using concrete homological constructions, unlike the existing $\infty$-categorical treatments which only give abstract answers. Our construction makes it possible to directly apply classical, homological techniques like Koszul duality.