nLab
homotopy limit

Idea

In great generality, a homotopy limit is a way of constructing appropriate sorts of limits in a (weak) higher category using some “presentation” of that higher category by a stricter structure. The general study of such presentations is homotopy theory.

In classical homotopy theory, the presentation is given by a category with weak equivalences, possibly satisfying extra axioms such as those of a homotopical category, a category of fibrant objects, or a model category. Such structures are considered to present an $(\mathrm{\infty }\mathrm{,1})$-category, and homotopy limits give a way of constructing the appropriate sort of ”$(\mathrm{\infty }\mathrm{,1})$-limits” in an $(\mathrm{\infty }\mathrm{,1})$-category.

In enriched homotopy theory, the presentation is given by an enriched model category or an enriched homotopical category, and it presents an “enriched $(\mathrm{\infty }\mathrm{,1})$-category.” Here the appropriate notion is a weighted homotopy limit, which is expected to construct “weighted $(\mathrm{\infty }\mathrm{,1})$-limits” in the presented “enriched $(\mathrm{\infty }\mathrm{,1})$-category.” Note that as yet, no fully general notion of “enriched $(\mathrm{\infty }\mathrm{,1})$-category” exists; see homotopical enrichment.

More concretely, in a simplicial model category that presents an (∞,1)-category, homotopy limits model the more intrinsic notion of limit in a quasi-category.

Definitions

As for ordinary limits, there are two ways to define homotopy limits:

• with explicit constructions that satisfy a local universal property: the homotopy limit object “represents homotopy-coherent cones up to homotopy.”

• as derived functors that satisfy a global universal property: the homotopy limit functor is “universal among homotopical approximations to the strict limit functor.”

One of the central theorems of the subject is that in good cases, the two give equivalent results; see below.

Global definition

Let $C$ be a category with weak equivalences and let $D$ be a (small) category. Make the functor category $[D,C]$ into a category with weak equivalences by taking the weak equivalences to be those natural transformations which are objectwise weak equivalences in $C$.

The homotopy limit of a functor $F:D\to C$ is the image of $F$ under the right derived functor, if it exists, of the limit functor ${\lim }_D:[D,C]\to C$ with respect to the given weak equivalences on $C$ and the objectwise weak equivalences on $[D,C]$:

$${\mathrm{holim}}_{D}F:=(R{\lim }_D)F.$$holim_D F := (R lim_D)F \,.

In the enriched case, this must be suitably modified to deal with weighted limits as well as enrichment of both $C$ and $D$.

Local definition

The local definition requires making precise the notion of a homotopy commutative cone over a diagram.

For the case of SimpSet-enrichment one elegant way to do so is in terms of suitable weighted limits as described in the example section at weighted limit: a homotopy commutative cone with tip $c\in C$ over a diagram $F:K\to C$ in an $SimpSet$-enriched category $C$ is a natural transformation $W\Rightarrow C(c,F(-)):K\to SimpSet$ where the weight functor $W$ is not constant on the point, as for ordinary limits, but is given by $W:k\mapsto N(K/k)$.

The same idea works if we are enriched over a category $V$ that is not $SimpSet$ but is itself enriched over $SimpSet$, such as topological spaces or spectra, since then any $V$-category becomes a $SimpSet$-category as well in a natural way. Finally, although a general model category need not be enriched over anything, it is always “almost” enriched over $SimpSet$, and so one can still make sense of this using the techniques of framings and resolutions; see the books of Hirschhorn and Hovey.

Following the reasoning described in Example 1 of representable functor one then defines the homotopy limit $L$ of a functor $F:K\to C$ to be a representing object for such homotopy cones, in the sense that we have a (weak) equivalence

$$\mathrm{Map}(X,L)\simeq \mathrm{HoCones}(X,F)$$Map(X,L) \simeq HoCones(X,F)

of hom-objects (spaces or simplicial sets in the classical context; enriched hom-objects in the enriched context).

Global versus local

The global definition is formulated in terms of weak equivalences only, while the local definition is formulated in terms of homotopies only. However, in practical cases, derived functors exist because their input objects (in this case, the diagram $F$) can be replaced by “good” (fibrant and/or cofibrant) objects in such a way that weak equivalences become homotopy equivalences. The derived functor of $\lim$ at the input object $F$ is then computed by applying the ordinary functor $\lim$ to a good replacement $RF$ of $F$.

It then turns out that the “good” (precisely, “fibrant”) replacement $RF$ “builds in” precisely the right homotopies so that an ordinary cone over $RF$ is the same as a homotopy-commutative cone over $F$. Therefore, $\lim (RF)$, which is the global homotopy-limit of $F$, is a representing object for homotopy-commutative cones over $F$, and thus is also a local homotopy-limit of $F$. There is a dual argument for colimits using cofibrant replacements.

Formal versions of this argument can be found in many places. Perhaps the original statement can be found in XI.8.1 of:

• A. K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations.

(As was often the case with Kan’s papers at that time, there are some details omitted from that treatment, but most are, as he claimed, quite easy to complete.) For another approach in an algebraic context, there is a description in Illusie’s thesis.

An abstract version in modern language, with proof, can be found in

• Michael Shulman, Homotopy limits and colimits and enriched homotopy theory.

Strictness

Another notable difference between the local and global definitions is that the global definition can only ever define the homotopy limit up to weak equivalence (isomorphism in the homotopy category), while in the local definition we could require, if we wanted to, an actual isomorphism

$$\mathrm{Map}(X,L)\cong \mathrm{HoCones}(X,F)$$Map(X,L) \cong HoCones(X,F)

of hom-objects, rather than merely a weak equivalence. By analogy with strict 2-limits, we may call such an object a strict homotopy limit.

Frequently a strict homotopy limit does in fact exist, and can be constructed as a weighted limit in the ordinary (enriched) category in question. In such cases, the strict homotopy limit may be easier to compute with than an arbitrary homotopy limit merely known to have the up-to-weak-equivalence universal property. Thus, sometimes when people say the homotopy limit they refer mean a strict homotopy limit.

When a strict homotopy limit exists, an arbitrary homotopy limit may be defined as an object which is (weakly) equivalent to the strict homotopy limit.

Homotopy limits versus higher categorical limits

As a special case of enriched homotopy theory, we may consider model categories or homotopical categories that are enriched over a notion of $n$-category as presentations for $(n+1)$-categories. (Here we allow $n$ to also be of the form (n,r), with the obvious convention that $(n,r)+1=(n+1,r+1)$ and $\mathrm{\infty }+1=\mathrm{\infty }$.) For example:

• simplicial sets are models for $\mathrm{\infty }$-groupoids ($(\mathrm{\infty }\mathrm{,0})$-categories), so simplicial model categories are presentations for $(\mathrm{\infty }\mathrm{,1})$-categories. Of course, arbitrary model categories are also presentations for $(\mathrm{\infty }\mathrm{,1})$-categories, but simplicial model categories are easier to work with, and in particular easier to construct homotopy limits in.

• A Cat-enriched category is just a strict 2-category, so a model 2-category or homotopical 2-category? is a presentation of a weak 2-category (or bicategory).

• A $(2\mathrm{Cat},{\otimes }_{\mathrm{Gray}}$)-enriched category is a Gray-category, and a model Gray-category? or homotopical Gray-category? is a presentation of a weak 3-category (or tricategory).

If $C$ is a category enriched over $(n-1)$-categories and we are considering it to be an $n$-category (which happens to be strict at the bottom level), it is natural to define a “weak equivalence” in the underlying ordinary category of $C$ to be a morphism that is an $n$-category-theoretic equivalence. We call this the natural or trivial homotopical structure on $C$. In certain cases (such as $n=2)$ it can be made into a model structure, also called natural or trivial.

Since higher categorical limits are generally defined as representing objects for cones that commute up to equivalence, it is unsurprising that if $C$ has a natural homotopical structure, locally-defined homotopy limits and $n$-limits coincide. For $n=1$ this is trivial. For $n=2$ it is proven in the paper of Gambino referenced below (particularly section 6). For $n=(\mathrm{\infty }\mathrm{,1})$ it is proven in (among other places) Lurie’s book, section 4.2.4. The case $n=3$ ought to be approachable in theory, but doesn’t seem to have been done (probably partly because the general theory of 3-limits is fairly nonexistent).

On the other hand, we may also consider a category $C$ enriched over $n$-categories with a larger class of weak equivalences than just the $n$-categorical equivalences. Then $C$ presents an $n$-category (its “homotopy $n$-category”) obtained by formally turing these weak equivalences into $n$-categorical equivalences. Homotopy limits in $C$ with this homotopical structure should then present $n$-limits in its homotopy $n$-category. In the case $n=(\mathrm{\infty }\mathrm{,1})$ this is also essentially in Lurie’s book; for other values of $n$ it may not be in the literature.

Examples

General weighted colimit formula for homotopy colimits

Let

• $C$ be a combinatorial simplicial model category

• let $D$ be a small simplicially enriched category

  (possibly an ordinary locally small category regarded as a SSet-enriched category in the tautological way)

• and let $F:D\to C$ be an SSet-enriched functor.

There is a general formula for the homotopy colimit over $F$ in terms of a coend or weighted colimit in $C$, using the following ingredients:

Ingredients

for $C$ a combinatorial simplicial model category as above and for $D$ any simplicially enriched category there is the projective and the injective global model structure on functors on the enriched functor category $[D,C]$.

• In the projective model structure $[D,C{]}_{\mathrm{proj}}$ the fibrations and the local equivalences are objectwise those of $C$

• In the injective model structure $[D,C{]}_{\mathrm{inj}}$ the cofibrations and the local equivalences are objectwise those of $C$.

Each of these is itself a combinatorial simplicial model category, so in particular the small object argument applies in these using which one obtains cofibrant replacement functors

$$Q_{\mathrm{proj}}:[D,C]\to [D,C]$$Q_{proj} : [D, C] \to [D, C]

and

$$Q_{\mathrm{inj}}:[D,C]\to [D,C].$$Q_{inj} : [D, C] \to [D, C] \,.

That $C$ is a simplicial model category implies in particular that it is tensored over SSet and that the tensoring functor

$$\otimes :C\times \mathrm{SSet}\to C$$\otimes : C \times SSet \to C

is a left Quillen bifunctor. By the properties of Quillen bifunctors discussed there, it follows that the coends over the tensor in the form

$${\int }^D(-)\otimes (-):[D^{\mathrm{op}},\mathrm{SSet}{]}_{\mathrm{proj}}\times [D,C{]}_{\mathrm{inj}}\to C$$\int^D (-) \otimes (-) : [D^{op},SSet]_{proj} \times [D,C]_{inj} \to C

and in the form

$${\int }^D(-)\otimes (-):[D^{\mathrm{op}},\mathrm{SSet}{]}_{\mathrm{inj}}\times [D,C{]}_{\mathrm{proj}}\to C$$\int^D (-) \otimes (-) : [D^{op},SSet]_{inj} \times [D,C]_{proj} \to C

both themselves left Quillen bifunctors.

Write

$$*:D\to \mathrm{SSet}$${*} : D \to SSet

for the functor that sends everything to the identity on the singleton set. This is both the tensor unit in the monoidal category $[D,\mathrm{SSet}]$.

General formula

Theorem

With the above assumptions and ingredients, the homotopy colimit over $F:D\to C$ is given either by

$$\mathrm{hocolim}F={\int }^D{Q}_{\mathrm{proj}}(*)\otimes Q_{\mathrm{inj}}(F)$$hocolim F = \int^D Q_{proj}({*}) \otimes Q_{inj}(F)

or by

$$\mathrm{hocolim}F={\int }^D{Q}_{\mathrm{inj}}(*)\otimes Q_{\mathrm{proj}}(F).$$hocolim F = \int^D Q_{inj}({*}) \otimes Q_{proj}(F) \,.

Proof

This is disucssed for instance in section 4 of

• Nicola Gambino, Weighted limits in simplicial homotopy theory (pdf)

Examples

Homotopy colimits over simplicial diagrams

Let $D={\Delta }^{\mathrm{op}}$ be the opposite category of the simplex category.

Proposition

A cofibrant replacement of the terminal object $*$ in the projective global model structure on functors $[\Delta ,\mathrm{SSet}]$ is the functor that assigns to $[n]$ the nerve of opposite category of the undercategory of $\Delta$ under $[n]$

$$N(-/\Delta {)}^{\mathrm{op}}:\Delta \to \mathrm{SSet}.$$N(-/\Delta)^{op} : \Delta \to SSet \,.

Proof

For instance prop 14.8.8 in

• Hirschhorn, Model categories and their localization

Notice that if $F:{\Delta }^{\mathrm{op}}\to C$ takes values in cofibrant objects of $C$, then it is itself cofibrant as an object of $[{\Delta }^{\mathrm{op}},C{]}_{\mathrm{inj}}$. In that case no further cofibrant replacement of $F$ is necessary and it therefore follows with the general formula and the above proposition that the homotopy colimit over $F$ is given by the formulas

$$\mathrm{hocolim}F={\int }^{[n]\in \Delta }N([n]/\Delta {)}^{\mathrm{op}}\otimes F(n).$$hocolim F = \int^{[n] \in \Delta} N([n]/\Delta)^{op} \otimes F(n) \,.

This is famously the formula introduced and used by Bousfield and Kan (but there originally missing the necessary condition that $F$ be objectwise cofibrant). See Bousfield-Kan map.

Homotopy pushouts

Let in the above general formula $D=\{a\leftarrow c\mathrm{to}b\}$ be the walking span. Ordinary colimits parameterized by such $D$ are pushouts. Homotopy colimits over such $D$ are homotopy pushouts.

In this simple case, we have the following simple observation:

Observation

For $D$ as above, the terminal functor $*:D\to \mathrm{SSet}$ is already cofibrant in $[D,\mathrm{SSet}{]}_{\mathrm{inj}}$.

Moreover

Observation

For $D$ as above, a functor $F:D\to C$ is cofibrant in $[D,C{]}_{\mathrm{proj}}$ if

• it sends both morphisms $c\to a$ and $c\to b$ to cofibrations

• it sends $c$ (and hence also $a$ and $b$) to cofibrant objects in $C$.

Since a coend $\int *\otimes F$ over a tensor product where the first factor in the integrand in the tensor unit is just an ordinary colimit over the remaining $F$, it follows that if $F$ is of the form of the above observation, then the ordinary colimit over $F$ already computes the homotopy pushout:

$$\mathrm{hocolim}F=\underrightarrow{\lim }F.$$hocolim F = \lim_\to F \,.

The dual version of this statement (for homotopy limits and homotopy pullbacks) is discussed in more detail in the examples below.

Homotopy pullbacks

Here we consider special cases of homotopy pullback in more detail.

Let $D=\{1\to 0\leftarrow 2\}$ be the pullback diagram, so that limits over it compute pullbacks, and assume that $F:D\to C$ is such that

$$F(1)\to F(0)\leftarrow F(2)$$F(1) \to F(0) \leftarrow F(2)

satisfies

• $F(i)$ is fibrant for all $i$;
• and either $F(1)\to F(0)$ or $F(2)\to F(0)$ is a fibration;

then

• ${\mathrm{holim}}_{D}F$ exists;
• and is weakly equivalent to the ordinary limit ${\mathrm{holim}}_{D}F\stackrel{\simeq }{\to }{\lim }_{D}F$.

Conversely this means that on an arbitrary pullback diagram ${\mathrm{holim}}_{D}F$ can be computed by finding a natural transformation $F\Rightarrow RF$ whose component morphisms are weak equivalences and such that $RF$ satisfies the above conditions.

Based loop objects

For $B$ any pointed object with point $\mathrm{pt}\stackrel{{\mathrm{pt}}_B}{\to }B$ the homotopy pullback of the point along itself is the loop space object of $B$

$$\begin{array}{ccc}{\Omega }_{\mathrm{pt}}& \to & *\\ \downarrow & & \downarrow \\ *& \to & B\end{array},$$\array{ \Omega_{pt} &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& B } \,,

i.e.

$$\mathrm{holim}(\mathrm{pt}\stackrel{{\mathrm{pt}}_B}{\to }B\stackrel{{\mathrm{pt}}_B}{\leftarrow }\mathrm{pt})\stackrel{\simeq }{\to }{\Omega }_{\mathrm{pt}}B.$$holim( pt \stackrel{pt_B}{\to} B \stackrel{pt_B}{\leftarrow} pt) \;\stackrel{\simeq}{\to}\; \Omega_{pt} B \,.

One way to compute this using the above prescription by noticing that the generalized universal bundle $E_{\mathrm{pt}}B$ provides a fibrant replacement of the pullback diagram in that we have

$$\begin{array}{ccccc}\mathrm{pt}& \to & B& \leftarrow & \mathrm{pt}\\ {\downarrow }^{\simeq }& & {\downarrow }^{\mathrm{Id}}& & {\downarrow }^{\mathrm{Id}}\\ E_{\mathrm{pt}}B& \to & B& \leftarrow & \mathrm{pt}\end{array}$$\array{ pt &\to& B &\leftarrow& pt \\ \downarrow^{\simeq} && \downarrow^{Id} && \downarrow^{Id} \\ \mathbf{E}_{pt}B &\to& B &\leftarrow& pt }

with all vertical morphisms weak equivalences and with the left bottom horizontal morphism a fibration.

More on that in the further examples below.

Fibration sequences

If $C$ is a pointed object, with point $*\to C$, then for a homotopy pullback of the form

$$\begin{array}{ccc}A& \to & *\\ \downarrow & & \downarrow \\ B& \to & C\end{array}$$\array{ A &\to& {*} \\ \downarrow && \downarrow \\ B &\to& C }

the sequence $A\to B\to C$ is called a fibration sequence. The object $A$ is the homotopy kernel or homotopy fiber of $B\to C$. Since homotopy pullback squares compose to homotopy pullback squares, the homotopy kernel of a homotopy kernel is not trivial, but is a loop space object

$$\begin{array}{ccccc}\Omega C& \to & A& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & B& \to & C\end{array}.$$\array{ \Omega C &\to& A &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& B &\to& C } \,.

Homotopy pullback of a point over a group / universal bundles

As a special case of the above general example we get the following.

Let $C=$ Grpd equipped with the folk model structure. Write $G$ for a group regarded as a discrete monoidal groupoid (elements of $G$ are the objects of the groupoids and all morphisms are identities) write and $BG$ for the corresponding one-object groupoid (single object, one morphism per element of $G$). Write $\mathrm{pt}$ for the terminal groupoid (one object, no nontrivial morphism). Notice that there is a unique functor $\mathrm{pt}\to BG$. Then we have

$$\mathrm{holim}\left(\begin{array}{ccc}& & \mathrm{pt}\\ & & \downarrow \\ \mathrm{pt}& \to & BG\end{array}\right)\stackrel{\simeq }{\to }G.$$holim \left( \array{ && pt \\ && \downarrow \\ pt &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} G \,.

To see this, we compute using the above prescription by finding a weakly equivalent pullback diagram such that one of its morphisms is a fibration. This is achived in particular by the generalized universal bundle $\mathrm{pt}\stackrel{\simeq }{\to }EG\to >BG$, where $EG$ is the action groupoid $G//G$ of $G$ acting on itself by multiplication from one side. So we have a weak equivalence of pullback diagrams

$$\begin{array}{ccccc}\mathrm{pt}& \to & BG& \leftarrow & \mathrm{pt}\\ {\downarrow }^{\simeq }& & {\downarrow }^{=}& & {\downarrow }^{=}\\ EG& \to & BG& \leftarrow & \mathrm{pt}\end{array}$$\array{ pt &\to& \mathbf{B}G &\leftarrow& pt \\ \downarrow^{\simeq} && \downarrow^= && \downarrow^= \\ \mathbf{E}G &\to& \mathbf{B}G &\leftarrow& pt }

and the homotopy limit in question is weakly equivalent to the ordinary limit over the lower diagram. That is directly seen to be $\mathrm{Disc}(\mathrm{Obj}(EG))=\mathrm{Disc}(\mathrm{Obj}(G//G))=\mathrm{Disc}(G)$ which we just write as $G$:

$$\mathrm{holim}\left(\begin{array}{ccc}& & \mathrm{pt}\\ & & \downarrow \\ \mathrm{pt}& \to & BG\end{array}\right)\stackrel{\simeq }{\to }\lim \left(\begin{array}{ccc}& & \mathrm{pt}\\ & & \downarrow \\ EG& \to & BG\end{array}\right)=G.$$holim \left( \array{ && pt \\ && \downarrow \\ pt &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} lim \left( \array{ && pt \\ && \downarrow \\ \mathbf{E}G &\to& \mathbf{B}G } \right) = G \,.

This example is important in the context of groupoidification and geometric function theory, as described there. A closely related example is the following: a functor $\rho :BG\to \mathrm{Top}$ is the datum of a toplogical space $X$ equipped with an action of $G$. Then, $\mathrm{colim}(\rho )=X/G$ whereas $\mathrm{hocolim}(\rho )=EG{\times }_{G}X$, see equivariant cohomology.

Homotopy pullback of a subgroup over a group

The above example generalizes straightforwardly to the case where the trivial inclusion $\mathrm{pt}\to BG$ is replaced by any inclusion $BH\hookrightarrow BG$ of any subgroup $H$ of $G$ pretty much literally by replacing $\mathrm{pt}$ by $BH$ throughout.

One finds

$$\mathrm{holim}\left(\begin{array}{ccc}& & BH\\ & & \downarrow \\ BH& \to & BG\end{array}\right)\stackrel{\simeq }{\to }H\backslash \backslash G//H$$holim\left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} H \backslash\backslash G//H

where on the right we have the action groupoid of $H\times H$ acting on $G$ by multiplication from the left (first factor) and the right (second factor).

To see this, we again build a fibrant replacement of the pullback diagram. Following the constructions at generalized universal bundle consider first the groupoid $E_{BH}G$ given by the pullback diagram

$$\begin{array}{ccc}{E}_{BH}G& \to & BH\\ \downarrow & & \downarrow \\ [I,BG]& \stackrel{d_0}{\to }& BG\\ {\downarrow }^{d_1}\\ BG\end{array}.$$\array{ \mathbf{E}_{\mathbf{B}H}G &\to& \mathbf{B}H \\ \downarrow && \downarrow \\ [I, \mathbf{B}G] &\stackrel{d_0}{\to}& \mathbf{B}G \\ \downarrow^{d_1} \\ \mathbf{B}G } \,.

As at generalized universal bundle one proves that the left vertical morphism $E_{BH}G\to BG$ is a fibration.

Now, notice (which was implicit in the above example) that since $[I,BG]$ is a path object in a category of fibrant objects we have a section $BG{\stackrel{\simeq }{\to }}^{\sigma }[I,BG]$ of $[I,BG]\stackrel{d_0}{\to }BG$. In the above pullback diagram this induces a morphism $BH\stackrel{\sigma }{\to }{E}_{BH}G$ making the obvious diagram commute. Now, the latter morphism, being the pullback of an acyclic fibration is an acyclic fibration, so its right inverse $\sigma$ is a weak equivalence. This way we obtain the morphism of pullback diagrams

$$\begin{array}{ccccc}BH& \to & BG& \leftarrow & BH\\ {}^{\simeq }{\downarrow }^{\sigma }& & {\downarrow }^{\mathrm{Id}}& & {\downarrow }^{\mathrm{Id}}\\ E_{BH}G& \to >& BG& \leftarrow & BH\end{array}$$\array{ \mathbf{B}H &\to& \mathbf{B}G &\leftarrow& \mathbf{B}H \\ {}^\simeq\downarrow^{\sigma} && \downarrow^{Id} && \downarrow^{Id} \\ \mathbf{E}_{\mathbf{B}H}G &\to \gt& \mathbf{B}G &\leftarrow& \mathbf{B}H }

which is objectwise a weak equivalence and such that the horizontal morphism on the bottom left is a fibration. By the above statement the ordinary limit of the lower horizontal diagram is weakly equivalent to the homotopy limit we are looking for. But this is manifestly the desired action groupoid:

$$\mathrm{holim}\left(\begin{array}{ccc}& & BH\\ & & \downarrow \\ BH& \to & BG\end{array}\right)\stackrel{\simeq }{\to }\lim \left(\begin{array}{ccc}& & BH\\ & & \downarrow \\ E_{BH}G& \to & BG\end{array}\right)\simeq \lim \left(\begin{array}{ccc}& & [I,BG]\\ & & {\downarrow }^{d_0\times d_1}\\ BH\times BH& \to & BG\times BG\end{array}\right)=H\backslash \backslash G//H.$$holim\left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} \lim\left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{E}_{\mathbf{B}H} G &\to& \mathbf{B}G } \right) \simeq \lim\left( \array{ && [I,\mathbf{B}G] \\ && \downarrow^{d_0 \times d_1} \\ \mathbf{B}H \times \mathbf{B}H &\to& \mathbf{B}G \times \mathbf{B}G } \right) = H \backslash\backslash G//H \,.

This example, too, is important at geometric function theory.

Homotopy span traces

• see the homotopy span traces discussed at span trace for more examples of homotopy pullbacks

Descent objects

Descent objects as they appear in descent and codescent are naturally conceived as homotopy limits. See also infinity-stack.

Homotopy (co)limits of simplicial pre(sheaves)

The local model structure on simplicial presheaves $\mathrm{SPSh}(C{)}_{\mathrm{proj}/\mathrm{inj}}^{\mathrm{loc}}$ over a site $C$ serve as models for ∞-stack (∞,1)-toposes.

Here we discuss some properties of homotopy limits and colimits in such model categories of simplicial presheaves.

Preservation of homotopy pullback by inverse images

For $C,C\prime$ two sites, a geometric morphism $p:\mathrm{Sh}(C)\overleftarrow{\to }\mathrm{Sh}(C\prime )$ of sheaf toposes induces correspondingly an adjunction

$$p:\mathrm{SSh}(C)\overleftarrow{\to }\mathrm{SSh}(C\prime ):p^{*}$$p : SSh(C) \stackrel{\leftarrow}{\to} SSh(C') : p^*

of simplicial (pre)sheaves. One would like this to extend to a Quillen adjunction that recalls the fact that it came from a geometric morphism by the fact that the left adjoint inverse image functor $\mathrm{SSh}(C\prime )\to \mathrm{SSh}(C)$ preserves finite homotopy limits.

In particular, if $C$ and $C\prime$ have the same underlying category but $C\prime$ the trivial coverage, then the geometric morphism in question is the inclusion of a reflective subcategory which typically induces a Bousfield localization of model categories that models the injection of a reflective (∞,1)-subcategory of ∞-stacks into $\mathrm{\infty }$-presheaves. Here the morphism $\mathrm{SPSh}(C\prime )\to \mathrm{SPSh}(C)$ is $\mathrm{\infty }$-stackification and should preserve finite homotopy limits.

The following result says that a strong version of this statement is true, at least for the preservation of homotopy pullbacks.

Theorem

Let $p:\mathrm{Sh}(C)\to \mathrm{Sh}(C\prime )$ be a geometric morphism of Grothendieck toposes. Let $p^{*}:\mathrm{Sh}(C\prime )\to \mathrm{Sh}(C)$ be the corresponding inverse image functor and let $s{p}^{*}:\mathrm{SSh}(C\prime )\to \mathrm{SSh}(C)$ be its degreewise extension to functor of simplicial sheaf categories.

Regarded as a functor between the corresponding local injective model structures on simplicial sheaves on both sides

$$s{p}^{*}:\mathrm{SSh}(C\prime {)}_{\mathrm{inj}}^{\mathrm{loc}}\to \mathrm{SSh}(C{)}_{\mathrm{inj}}^{\mathrm{loc}}$$s p^* : SSh(C')_{inj}^{loc} \to SSh(C)_{inj}^{loc}

this functor preserves homotopy pullbacks.

Proof

This appears as theorem 1.5 in

• Charles Rezk, Fibrations and homotopy colimits of simplicial sheaves (arXiv)

References

• A. K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations. Springer LNM 304. The classical reference; see in particular chapter XI.

• Wojciech Chacholski and Jerome Scherer, Homotopy theory of diagrams (arXiv) includes a global definition of homotopy (co)limit as 4.1, p. 14, and discusses how to compute them (co)limits concretely using local constructions. For instance the above statement on the computation of homotopy pullbacks is proposition 2.5, p. 15

• Philip Hirschhorn, Model categories and their localizations. Defines and studies (local) homotopy limits in model categories.

• Dwyer, Hirschhorn, Kan, Smith, Homotopy limit functors in model categories and homotopical categories. Defines global homotopy limits in homotopical categories and computes them using local constructions.

• Michael Shulman, Homotopy limits and colimits and enriched homotopy theory. Constructs and compares local and global weighted homotopy limits in enriched homotopical categories.

• Nicola Gambino, Homotopy limits for 2-categories (pdf), published as: Mathematical Proceedings of the Cambridge Philosophical Society 145 (2008) 43-63.) Proves that homotopy limits in a 2-category with its natural model structure coincide with 2-categorical pseudo-limits, and hence give 2-limits.

• Jacob Lurie, Higher Topos Theory. Lots of stuff about $(\mathrm{\infty }\mathrm{,1})$-categories, including the computation of homotopy limits (section 4.2.4).

• Andre Hirschowitz, Carlos Simpson. Descent pour les n-champs. Probably there is some good stuff in here about homotopy limits and limits in $(\mathrm{\infty },n)$-categories.

Tim Mike, are you intending to treat the case of when the domain category, $D$ is the above, is enriched as well? This would handle the example of homotopy limts of homotopy coherent diagrams, both in Vogt’s sense and in the simplicially enriched case looked at by Bourn and Cordier. This would also allow the $G$ in one of the examples to be a simplicial or topological group, or to be (?) and A-infinity category. (Some of those examples may be already dealt with in others of the entries as different people classify things in different ways.)

Perhaps some of the more classical referencs, Vogt, Bousfield-Kan etc. might be included for completeness.

Mike: Yes, certainly; my paper referenced above deals with the case when $D$ is enriched as well. There are cofibrancy technicalities, of course. I’m not against including the classical references, although I find them fairly impenetrable myself.