nLab homotopy limit

Idea

In great generality, a homotopy limit is a way of constructing an appropriate sorts of limits in a (weak) higher category and in general and in (∞,1)-category theory in particular, using some presentation of that higher category by a 1-categorical 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 (∞,1)-category, and homotopy limits give a way of constructing the appropriate sort of (∞,1)-categorical limits.

In enriched homotopy theory, the presentation is given by an enriched model category or an enriched homotopical category, and it may presents an “enriched $(\infty,1)$-category” or be a more powerful presentation of a bare $(\infty,1)$-category (notably if the enrichment is over sSet). In the enriched category theory context the appropriate notion is a weighted homotopy limit, which may construct “weighted $(\infty,1)$-limits” in the presented “enriched $(\infty,1)$-category” or may be a more powerful tool for constructing plain $(\infty,1)$-categorical limits (in particular if the enrichment is over sSet). Note that as yet, no fully general notion of “enriched $(\infty,1)$-category” exists; see homotopical enrichment.

Maybe the most commonly encountered setup for homotopy limits is that where the (∞,1)-category in question is presented by a simplicial model category. See also homotopy Kan extension, of which (globally defined) homotopy (co)limits are a special case.

Terminology

In older literature, such as the original work of Bousfield–Kan, the homotopy limit of a diagram $X\colon I\to sSet$ is defined as the weighted limit

$$hom(I/-, X),$$

where the weight $I/-$ sends $i\mapsto N(I/i)$, the nerve of the comma category of $i$ (for more see the section “Resolved (co)ends” Bousfield-Kan map).

Such a functor must be derived in order to get the correct (homotopy invariant) notion, which in this case amounts to replacing $X$ with an objectwise weakly equivalent diagram valued in Kan complexes.

At some point (when?) a shift in terminology happened, and in the modern parlance homotopy limits are commonly assumed to be derived.

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 of a homotopy Kan extension 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) diagram 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 ordinary limit and colimit operations on $D$-diagrams are (as described there) the right and left adjoints of the functor $const : C \to [D,C]$, or equivalently left and right Kan extension along the unique functor $!\colon D\to *$ to the terminal category.

$$[D,C] \stackrel{\stackrel{\overset{colim = Lan_!}{\longrightarrow}}{\overset{const}{\leftarrow}}}{\underset{lim = Ran_!}{\to}} C \,.$$

The (globally defined) homotopy limit and colimit are accordingly the right and left homotopy Kan extension along $!\colon D\to *$:

• The homotopy limit of a functor $F : D \to C$ is, if it exists, the image of $F$ under the right derived functor 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 $[D,C]$:

  $$holim_D F := (\mathbb{R} lim_D)F \,.$$

• The homotopy colimit of a functor $F : D \to C$ is, if it exists, the image of $F$ under the left derived functor of the colimit functor $colim_D : [D,C] \to C$ with respect to the given weak equivalences on $C$ and the objectwise weak equivalences on $[D,C]$:

$$hocolim_D F := (\mathbb{L} colim_D)F \,.$$

Alternative definitions can be formulated at the level of the homotopy category $W^{-1} C$ one defines a localized version $\bar{\Delta}^I : W^{-1} C\to W_I^{-1} C^I$ of the diagonal functor $\Delta^I : C\to C^I$ and define the homotopy limits and colimits as the adjoints of $\bar{\Delta}^I$ (at least at the points where the adjoints are defined). Here $W_I\subset Mor(C^I)$ are the morphisms of diagrams whose all components are in $W\subset Mor(C)$. The above definitions via derived functors (Kan extensions) follow once one applies the general theorem that the derived functors of a pair of adjoint functors are also adjoint and noticing that $(\Delta^I,\bar{\Delta}^I)$ is a morphism of localizers (and in particular that $\bar{\Delta}^I$ with the identity 2-cell is a Kan extension (simultaneously left and right)).

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

In particular, if $C$ is equipped with the extra structure of a simplicial model category and $D$ is an (small) sSet-enriched category we may also hope to equip the enriched functor category $[D,C]$ with the structure of a simplicial model category. There are two different “canonical” such structures:

• the projective model structure on functors $[D,C]_{proj}$;

• the injective model structure on functors $[D,C]_{inj}$

(both of which have the same objectwise weak equivalences and are in fact Quillen equivalent). When these model structures exist (as they do when $C$ is combinatorial), limit and colimit constitute then two $sSet$-enriched Quillen adjunctions

$$[D,C]_{proj} \stackrel{\overset{colim}{\longrightarrow}}{\underset{const}{\longleftarrow}} C$$

and

$$[D,C]_{inj} \stackrel{\overset{const}{\longleftarrow}}{\underset{lim}{\longrightarrow}} C \,.$$

(All proofs and other technical details on this are at homotopy Kan extension.)

These present directly the corresponding adjoint (∞,1)-functors (as described there)

$$(\infty,1)Func(D,C^\circ) \stackrel{\longrightarrow}{\stackrel{\leftarrow}{\longrightarrow}} C^\circ$$

by precomposition with a cofibrant replacement functor (for the colimit) and a fibrant replacement functor (for the limit).

Local definition

The local definition requires making precise the notion of a homotopy commutative cone on 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 $c \in C$ on a diagram $F : K \to C$ in an $\Simp\Set$-enriched category $C$ is a natural transformation $W \Rightarrow C(c,F(-)) : K \to \Simp\Set$ 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 $\Simp\Set$, such as topological spaces or spectra, since then any $V$-category becomes a $\Simp\Set$-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 $\Simp\Set$, 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

$$Map(X,L) \simeq HoCones(X,F)$$

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 $R F$ of $F$.

It then turns out that the “good” (precisely, “fibrant”) replacement $R F$ “builds in” precisely the right homotopies so that an ordinary cone on $R F$ is the same as a homotopy-commutative cone on $F$. Therefore, $lim (R F)$, which is the global homotopy-limit of $F$, is a representing object for homotopy-commutative cones on $F$, 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

$$Map(X,L) \cong HoCones(X,F)$$

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.

Derivators

It is noteworthy that the homotopy limit and colimit in a category with weak equivalences are drastically different from the ordinary limit and colimit in the corresponding homotopy category: the universal property of the full $(\infty,1)$-categorical limits and colimits crucially does take into account the explicit higher cells and does not work just up to any higher cells.

This (obvious) observation is a very old one and can be seen to be one of the driving forces behind the insight that just working with homotopy categories misses crucial information, something which today is understood as the statement that a homotopy category is just the decategorification of an (∞,1)-category.

While the full theory of (∞,1)-categories is one way to impose the correct notion of higher categorical limits, there are other ways to deal with issue. Due to Alexander Grothendieck is the technique of using derivators for keeping track of homotopy limits.

Roughly, the idea of a derivator is that while the single homotopy category $Ho(C)$ of a category $C$ with weak equivalences is not sufficient information, the homotopy limit of a diagram in $[D,C]$ is encoded in the homotopy category $Ho([D,C])$ of that functor category (this is after all the domain of the plain 1-categorical derived functor of the limit functor). Accordingly, what is called the derivator of a category with weak equivalences $C$ is a collection of all the homotopy categories $Ho([D,C])$ of all the possible diagram categories $[D,C]$, as $D$ runs over all small categories. See there for more details.

Computational methods

Above we defined homotopy (co)limits in general. There are various more specific formulas and algorithms for computing homotopy (co)limits. Here we discuss some of these

Ordinary (co)limits on resolved diagrams

The direct prescription for computing the value of a right or left derived functor between model categories is by evaluating the original functor on a fibrant or cofibrant resolution of the given object.

For the derived functors of limit and colimit

$$[D,C] \stackrel{\overset{\lim_\longleftarrow}{\longrightarrow}}{\stackrel{\overset{const}{\longleftarrow}}{\underset{\lim_\longrightarrow}{\longrightarrow}}} C$$

let for instance

$Q_{proj} : [D,C] \to [D,C]$ be a cofibrant replacement for the projective model structure on functors, so that for any diagram $F$ the diagram $Q_{proj} F$ is a projectively cofibrant diagram (see there for more details). Then the homotopy colimit is presented by the ordinary colimit on $Q_{proj} F$:

$$(\mathbb{L}\underset{\to}{\lim}) F \simeq \underset{\to}{\limt} Q_{proj}(F) \,.$$

This is sometimes called the Quillen formula for computing homotopy colimits. Analogously with $P_{inj}$ a fibrant replacement functor for the injective model structure, we have

$$holim F \simeq lim P_{inj}(F) \,.$$

Often, however, it is inconvenient to produce a resolution of a diagram. Because often all the work is in finding the resolution, while it is easy to evaluate the original functor on it. Therefore one wants ways to slightly change the setup of the problem such that the computation of the resolutions becomes more systematic. One such way is the use of derived (co)ends, discussed below.

Resolved (co)ends

Let $Q'_{proj}$ a cofibrant replacement functor for $[D^{op}, sSet_{Quillen}]_{proj}$ (notice the opposite category) and $Q_{inj}$ one for $[D,C]_{inj}$. Let $* \in [D^op,sSet]$ simplicial presheaf constant on the terminal simplicial set and $Q'_{proj}(')$ its projective cofibrant replacement.

Then we also have the coend expression

$$hocolim F \simeq \int^{D} Q'_{proj}(*) \cdot Q_{inj}(F) \,,$$

where in the integrand we have the tensoring of the simplicial model category $C$ over sSet.

If $D$ happens to be a Reedy model category we can equivalently use in this expression also the Reedy model structures $[D,C]_{Reedy}$ and $[D^{op}, C]_{Reedy}$ and obtain the homotopy colimit as

$$hocolim F \simeq \int^{D} Q'_{Reedy}(*) \cdot Q_{Reedy}(F) \,,$$

This formula is sometimes called the Bousfield-Kan formula (see at Bousfield-Kan map). The coend is a weighted limit and this formula for the plain homotopy colimit can be understood the left derived weighted colimit which trivial weight (the underived weight is trivial, but becomes non-trivial after derivation – this extra complexity helps reduce the complexity for the replacement for the functor $F$ itself).

In detail, let $V$ be a monoidal model category and $C$ a $V$-enriched model category (for instance $V = sSet_{Quillen}$ as in the above discussion). Let $D$ be a small $V$-enriched category. Then for a any given weight

$$W : D^{op} \to V$$

we have a $V$-adjunction

$$[D,C] \underoverset {\underset{const(-)^{W}}{\longleftarrow}} {\overset{W \otimes_D (-)}{\longrightarrow}} {\;\;\;\bot\;\;\;} D \,,$$

where the left adjoint is the weighted colimit given by the coend

$$W \otimes_D F = \int^{d \in D} W(d) \otimes F(d)$$

(with the tensoring over $V$ in the integran) and where the right adjoint is formed using the cotensor over $V$.

The crux now is that as $W$ varies, the left adjoint here is a left Quillen bifunctor

$$\int (-)\cdot (-) : [D^{op}, V] \times [D,C] \to C \,.$$

For details on this see Quillen bifunctor or around page 9 of (Gambino 10).

From the fact that this is a Quillen bifunctor and using the observation that for the trivial weight $W = const 1$ the weighted colimit reduces to an ordinary colimit, follows the above Bousfield-Kan-type formula for the homotopy colimit.

Bar constructions

A general way of obtaining resolutions that compute homotopy (co)limits involves bar constructions. (…)

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 $\infty+1=\infty$.) For example:

• simplicial sets are models for $\infty$-groupoids ($(\infty,0)$-categories), so simplicial model categories are presentations for $(\infty,1)$-categories. Of course, arbitrary model categories are also presentations for $(\infty,1)$-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 $(2Cat,\otimes_{Gray}$)-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 (Gambino 10) (particularly section 6). For $n=(\infty,1)$ 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 $n=(\infty,1)$ this is also essentially in Lurie’s book; for other values of $n$ it may not be in the literature.

Homotopy limits versus limits in the homotopy category

It is important to note that homotopy limits and limits in the homotopy category are, in general, incomparable. A homotopy limit need not be a limit in the homotopy category, while a limit in the homotopy category need not be a homotopy limit.

It is generally true that homotopy products (and coproducts) are also products and coproducts in the homotopy category. Some other homotopy limits induce the corresponding notion of weak limit in the homotopy category; for instance, homotopy pullbacks become weak pullbacks in the homotopy category. However, even this is not true for all types of homotopy limit.

On the other hand, homotopy categories do not usually have many limits and colimits at all (aside from products and coproducts). An explicit proof that $Ho(Cat)$ does not have pullbacks can be found here. But even if a homotopy category does happen to have limits and colimits, these need not be the same as homotopy limits.

For instance, every chain complex over a field $k$ is quasi-isomorphic to its homology, regarded as a chain complex with zero differentials; and between chain complexes of the latter form, quasi-isomorphisms are just isomorphisms. Thus, the homotopy category of chain complexes over $k$ is equivalent to the category of graded $k$-vector-spaces. This is complete and cocomplete as a category, but its limits and colimits are not the same as the homotopy limits and colimits arising from its presentation as the homotopy category of chain complexes. In particular, chain complexes are a stable (infinity,1)-category, so every homotopy pullback square is also a homotopy pushout square and vice versa; but nothing of the sort is true in graded vector spaces as a 1-category.

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 $[D,C]_{proj}$ the fibrations and the local equivalences are objectwise those of $C$

• In the injective model structure $[D,C]_{inj}$ 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

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

and

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

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

$$\otimes : C \times sSet \to C$$

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

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

and in the form

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

both themselves left Quillen bifunctors.

Write

$${*} : D^{op} \to sSet$$

for the functor that sends everything to the identity on the singleton set. This is the tensor unit in the monoidal category $[D^{op},sSet]$.

General formula

Theorem

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

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

or by

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

Proof

By the fact that the coend over the tensor appearing here is a Quillen bifunctor.

This is disucssed for instance in section 4 of (Gambino 10).

Examples

Homotopy colimits of simplicial diagrams

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

Proposition

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

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

Proof

For instance prop 14.8.8 in

• Hirschhorn, Model categories and their localization

Notice that if $F : \Delta^{op} \to C$ takes values in cofibrant objects of $C$, then it is itself cofibrant as an object of $[\Delta^{op},C]_{inj}$. 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

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

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 $D = \{a \leftarrow c \to b\}$ 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 ${*} : D \to sSet$ is already cofibrant in $[D,sSet]_{inj}$.

Moreover

Observation

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

$$hocolim F = \lim_\to F \,.$$

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

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

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

• $holim_D F$ exists;
• and is weakly equivalent to the ordinary limit $holim_D F \stackrel{\simeq}{\longrightarrow} lim_D F$.

Conversely this means that on an arbitrary pullback diagram $holim_D F$ can be computed by finding a natural transformation $F \Rightarrow R F$ whose component morphisms are weak equivalences and such that $R F$ satisfies the above conditions.

Based loop objects

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

$$\array{ \Omega_{pt} &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& B } \,,$$

i.e.

$$holim( pt \stackrel{pt_B}{\longrightarrow} B \stackrel{pt_B}{\leftarrow} pt) \;\stackrel{\simeq}{\longrightarrow}\; \Omega_{pt} B \,.$$

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

$$\array{ pt &\to& B &\leftarrow& pt \\ \downarrow^{\simeq} && \downarrow^{Id} && \downarrow^{Id} \\ \mathbf{E}_{pt}B &\to& B &\leftarrow& pt }$$

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

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

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

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

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 canonical 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) and write $\mathbf{B}G$ for the corresponding one-object groupoid (single object, one morphism per element of $G$). Write $pt$ for the terminal groupoid (one object, no nontrivial morphism). Notice that there is a unique functor $pt \to \mathbf{B}G$. Then we have

$$holim \left( \array{ && pt \\ && \downarrow \\ pt &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\longrightarrow} G \,.$$

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 achieved in particular by the generalized universal bundle $pt \stackrel{\simeq}{\longrightarrow} \mathbf{E}G \to\gt \mathbf{B}G$, where $\mathbf{E}G$ 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

$$\array{ pt &\to& \mathbf{B}G &\leftarrow& pt \\ \downarrow^{\simeq} && \downarrow^= && \downarrow^= \\ \mathbf{E}G &\to& \mathbf{B}G &\leftarrow& pt }$$

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

$$holim \left( \array{ && pt \\ && \downarrow \\ pt &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\longrightarrow} lim \left( \array{ && pt \\ && \downarrow \\ \mathbf{E}G &\to& \mathbf{B}G } \right) = G \,.$$

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 $\rho:\mathbf{B}G\to {Top}$ is the datum of a toplogical space $X$ equipped with an action of $G$. Then, $colim(\rho)=X/G$ whereas $hocolim(\rho)=\mathbf{E}G\times_G X$, see equivariant cohomology.

Homotopy pullback of a subgroup over a group

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

One finds

$$holim\left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\longrightarrow} H \backslash\backslash G//H$$

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). (See for instance at Hecke category for an application.)

To see this, we again build a fibrant replacement of the pullback diagram. Following the constructions at generalized universal bundle consider first the groupoid $\mathbf{E}_{\mathbf{B}H}G$ given by the pullback diagram

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

As at generalized universal bundle one proves that the left vertical morphism $\mathbf{E}_{\mathbf{B}H}G \to \mathbf{B}G$ is a fibration.

Now, notice (which was implicit in the above example) that since $[I,\mathbf{B}G]$ is a path object in a category of fibrant objects we have a section $\mathbf{B}G \stackrel{\simeq}{\to}^\sigma [I, \mathbf{B}G]$ of $[I,\mathbf{B}G] \stackrel{d_0}{\to} \mathbf{B}G$. In the above pullback diagram this induces a morphism $\mathbf{B}H \stackrel{\sigma}{\to} \mathbf{E}_{\mathbf{B}H}G$ 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

$$\array{ \mathbf{B}H &\to& \mathbf{B}G &\leftarrow& \mathbf{B}H \\ {}^\simeq\downarrow^{\sigma} && \downarrow^{Id} && \downarrow^{Id} \\ \mathbf{E}_{\mathbf{B}H}G &\to \gt& \mathbf{B}G &\leftarrow& \mathbf{B}H }$$

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:

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

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

Homotopy colimits of simplicial diagrams

Recall that every simplicial set is the colimit of its elements (this Prop.). The following says that this is in fact a homotopy colimit, up to equivalence.

Proposition

(simplicial set is homotopy colimit of its cells)

Every simplicial set is the homotopy colimit over its cells.

Precisely: for $X \in$ sSet a simplicial set, let

$$\tilde X \;\colon\; \Delta^{op} \xrightarrow{\;} Set \xhookrightarrow{\;} sSet$$

be the corresponding bisimplicial set which in degree $k$ is the the constant simplicial set on the set $X_k$ of $k$-simplices.

For the standard homotopical structure on $sSet^{\Delta^{op}}$, the homotopy colimit over $\tilde X$ is equivalent to the origianal $X$:

$$hocolim \tilde X \stackrel{\simeq}{\to} X$$

in the standard model structure on simplicial sets.

Proof

Use the Reedy model structure $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$. With the coend recipe for the hocolim discussed above, it follows that the hocolim is the coend

$$\int^{[k] \in \Delta} Q'_{Reedy}(*) \times Q_{Reedy}(\tilde X) \,,$$

where $Q'_{Reedy}(\cdots)$ is a cofibrant resolution in the Reedy model structure $[\Delta,sSet_{Quillen}]_{Reedy}$ and $Q_{Reedy}(...)$ in $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$. But by the discussion at Reedy model structure – simplex category we have that

• $\tilde X$ is always cofibrant in $[\Delta^{op},sSet_{Quillen}]_{Reedy}$;

• a cofibrant resolution of the point in $[\Delta, sSet_{Quillen}]_{Reedy}$ is given by $\Delta[-] : \Delta \to sSet$.

It follows that the hocolim is given by

$$\int^{[k] \in \Delta} \Delta[k] \times X_k \;\simeq\; X \,.$$

which is the expression of $X$ is an ordinary colimit of its elements (this Prop., the “co-Yoneda lemma”).

Remark

More generally with this kind of argument it follows that generally the homotopy colimit over a simplicial diagram of simplicial sets is represented by the diagonal simplicial set of the corresponding bisimplicial set.

Remark

This kind of argument has many immediate generalizations. For instance for $C = [K^{op}, sSet_{Quillen}]_{inj}$ the injective model structure on simplicial presheaves over any small category $K$, or any of its left Bousfield localizations, we have that the cofibrations are objectwise those of simplicial sets, hence objectwise monomorphisms, hence it follows that every simplicial presheaf $X$ is the hocolim over its simplicial diagram of component presheaves.

For the following write $\mathbf{\Delta} : \Delta \to sSet$ for the fat simplex.

Lemma

The fat simplex is Reedy cofibrant.

Proof

By the discussion at homotopy colimit, the fat simplex is cofibrant in the projective model structure on functors $[\Delta, sSet_{Quillen}]_{proj}$. By the general properties of Reedy model structures, the identity functor $[\Delta, sSet_{Quillen}]_{proj} \to [\Delta, sSet_{Quillen}]_{Reedy}$ is a left Quillen functor, hence preserves cofibrant objects.

Corollary

For $X \in [\Delta^{op}, C]$ a Reedy cofibrant object, the Bousfield-Kan map

$$\int^{[n]} \mathbf{\Delta}[n] \cdot X_n \to \int^{[n]} \Delta[n] \cdot X_n$$

is a weak equivalence in $C$.

Proof

The coend over the tensor is a left Quillen bifunctor

$$\int (-)\cdot (-) : [\Delta,sSet_{Quillen}]_{Reedy} \times [\Delta^{op}, C]_{Reedy}$$

(as discussed there). Therefore with its second argument fixed and cofibrant it is a left Quillen functor in the remaining argument. As such, it preserves weak equivalences between cofibrant objects (by the factorization lemma). By the above discussion, both $\mathbf{\Delta}[n]$ and $\Delta[-]$ are indeed cofibrant in $[\Delta,sSet_{Quillen}]_{Reedy}$. Clearly the functor $\mathbf{\Delta}[-] \to \Delta[-]$ is objectwise a weak equivalence in $sSet_{Quillen}$, hence is a weak equivalence.

Proposition

Let $i : \Delta_f \hookrightarrow \Delta$ be the inclusion into the simplex category of all the monomorphisms (all the face maps).

This inclusion is a homotopy-initial functor. As a consequence, homotopy colimits of shape $\Delta$ can equivalently be computed after their restriction to $\Delta_f$

$$hocolim( \Delta^{op} \stackrel{F}{\to} C) \simeq hocolim( \Delta_f^{op} \stackrel{i^{op}}{\to} \Delta^{op} \stackrel{F}{\to} C) \,.$$

See Dugger, example 18.2.

Homotopy colimits of diagrams of spaces

By geometric realization

The following is sometimes in the literature taken as the definition of homotopy colimits of diagrams of spaces. It is one of the earliest formulas for there.

Definition

Let $D$ be a category and $F : D \to$ Top a functor.

Define a simplicial topological space $sF$ by setting

$$sF : [n] \mapsto \coprod_{d_0 \leftarrow d_1 \leftarrow \cdots \leftarrow d_n} F(d_n)$$

and using the obvious face and degeneracy maps: face maps act by mapping components of the coproducts of one sequence of morphisms to one obtained by deleting outer arrows or composing inner arrows. If the rightmost arrow is deleted, then the component map is not the identity but is $F(d_n) \to F(d_{n-1})$. The degeneracy maps similarly introduce identity morphisms.

Theorem

Let $D$ be a category and $F : D \to$ Top a functor. Then the homotopy colimit of $F$ is equivalent to the geometric realization of simplicial topological spaces of $s F$:

$$hocolim F \simeq \vert sF \vert \,.$$

This is an application of the bar-construction method.

See for instance Dugger, part 1 for an exposition.

Higher homotopy van Kampen theorem

Theorem

Let $X$ be a topological space, write $Op(X)$ for its category of open subsets and let

$$\chi : C \to Op(C)$$

a functor out of a small category $C$ such that

• for each point $x\in X$ the full subcategory $C_x$ of objects $c$ such that $\chi(x)$ contains $x$ has a weakly contractible nerve.

Then:

the canonical morphism in sSet out of the colimit

$${\lim_\to} Sing \circ \chi \to Sing(X)$$

into the singular simplicial complex of $X$ exhibits $Sing(X)$ as the homotopy colimit $hocolim Sing \circ \chi$.

See higher homotopy van Kampen theorem for details.

Sequential homotopy (co)limits

General

For

$$D = \{0 \to 1 \to 2 \to \cdots\}$$

the cotower category, a colimit of shape $D$ is called a sequential colimit. For $C$ a combinatorial model category it is easy to characterize the cofibrant objects in the projective model structure on functors $[D, C]_{proj}$: these are those cotower diagrams all whose morphisms are cofibrations and whose 0th object (and hence all objects) are cofibrant.

So given a cotower with such a property, its homotopy colimit is just the ordinary sequential colimit in $C$.

Dually for sequential limits of a tower diagram.

A standard application for this is for instance the construction of the classifying space $B U = \underset{\underset{n}{\longrightarrow}}{\lim} B U(n)$ for reduced topological K-theory. See at stable unitary group for more.

$\lim^1$ and Milnor sequences

See at lim^1 and Milnor sequences

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 $SPSh(C)_{proj/inj}^{loc}$ 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 $C, C'$ two sites, a geometric morphism $p : Sh(C) \stackrel{\leftarrow}{\to} Sh(C')$ of sheaf toposes induces correspondingly an adjunction

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

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 $SSh(C') \to SSh(C)$ preserves finite homotopy limits.

In particular, if $C$ and $C'$ have the same underlying category but $C'$ 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 $\infty$-presheaves. Here the morphism $SPSh(C') \to SPSh(C)$ is $\infty$-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 $p : Sh(C) \to Sh(C')$ be a geometric morphism of Grothendieck toposes. Let $p^* : Sh(C') \to Sh(C)$ be the corresponding inverse image functor and let $s p^* : SSh(C') \to SSh(C)$ 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

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

this functor preserves homotopy pullbacks.

Proof

This appears as theorem 1.5 in

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

More examples and special cases

• filtered homotopy colimit

Related concepts

• (∞,1)-limitlax (∞,1)-colimit

• mapping cone, mapping telescope

References

General

Classical references:

• Aldridge K. Bousfield and Daniel M. Kan, Chapter IX in: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304 (1972; 1987), Springer (doi:10.1007/978-3-540-38117-4)

• Rainer Vogt, Homotopy limits and colimits, Math. Z. 134 (1973) 11-52 [eudml:171965, doi:10.1007/BF01219090]

See

• Frank Adams, part III, section 8 of Stable homotopy and generalised homology, 1974

More recently one has:

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113, AMS 2004 (ISBN: 978-1-4704-1340-8, pdf)

• Emily Riehl, Chapter 5 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

An introduction:

• Daniel Dugger, A primer on homotopy colimits, (2008) [pdf, pdf, pdf]

A general overview via universal properties is in the

• Georges Maltsiniotis lectures, Sevilla (2008)

  I, localizers, (pdf);

  II, model categories, (pdf);

  III, derivators, (pdf):

  IV, summary on derivators (pdf)

In

• Wojciech Chacholski, Jerome Scherer, Homotopy theory of diagrams (arXiv)

is given a global definition of homotopy (co)limit as 4.1, p. 14, and it is discussed how to compute homotopy (co)limits concretely using local constructions. For instance the above statement on the computation of homotopy pullbacks is proposition 2.5, p. 15

A nice discussion of the expression of homotopy colimits in terms of coends is in

• Nicola Gambino, Weighted limits in simplicial homotopy theory, Journal of Pure and Applied Algebra Volume 214, Issue 7, July 2010, Pages 1193–1199 (pdf, publisher).

A collection of examples and exercises is in

• Samuel Isaacson, Excercises in homotopy colimits (pdf)

See also

• Samuel Isaacson, A note on unenriched homotopy coends (pdf)

Homotopy limits for triangulated categories are studied in

• Marcel Bökstedt, Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209–234, MR94f:18008, numdam

Other references are

• 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, math.CT/0610194. Constructs and compares local and global weighted homotopy limits in enriched homotopical categories, where the domain category is also allowed to be enriched.

• Kensuke Arakawa, Homotopy Limits and Homotopy Colimits of Chain Complexes, arXiv:2310.00201. The case of chain complexes.

• Sergey Arkhipov, Sebastian Ørsted, Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories, arxiv. A version of Weighted limits in simplicial homotopy theory that doesn’t require simplicial enrichment.

• 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 $(\infty,1)$-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 $(\infty,n)$-categories.

• Beatriz Rodriguez Gonzalez, Realizable homotopy colimits (arXiv:1104.0646)

• MathOverflow question: universal-problem-that-motivates-the-definition-of-homotopy-limits

Discussion in the context of the (infinity,1)-Grothendieck construction is in

• Gijs Heuts, Ieke Moerdijk, Left fibrations and homotopy colimits (arXiv:1308.0704)

A discussion of homotopy limits via weighted limits may be found in:

• Dominique Bourn, and Jean-Marc Cordier. A general formulation of homotopy limits, Journal of Pure and Applied Algebra 29.2 (1983): 129-141.

In homotopy type theory

A formalization of some aspects of homotopy limits in terms of homotopy type theory is Coq-coded in

• Guillaume Brunerie, HoTT/Coq/Limits

• Jeremy Avigad, Chris Kapulkin, Peter LeFanu Lumsdaine, Homotopy limits in Coq (arXiv:1304.0680)