homotopy limit

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#### Homotopy theory
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#### Model category theory
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#### $(\infty,1)$-Limits and colimits
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#Contents#

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## Idea

In great generality, a _homotopy limit_ is a way of constructing an appropriate sorts of [[limit]]s in a [[higher category theory|(weak) higher category]] and in general and in [[(∞,1)-category]] theory in particular, using some _presentation_ of that higher category by a [[category theory|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 [[limit in a quasi-category|(∞,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 limit|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 [[presentable (∞,1)-category|presented]] by a [[simplicial model category]]. See also [[homotopy Kan extension]], of which (globally defined) homotopy (co)limits are a special case.

## Terminology
 {#Terminology}

In older literature, such as the original work of [Bousfield–Kan](#BousfieldKan), 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](#ResolvedCoEnds)" *[[Bousfield-Kan map]]*).

Such a functor must be [[derived functor|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 [[limit]]s,
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 functor]]s 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 {#GlobalDefinition}

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 transformation]]s which are objectwise weak equivalences in $C$.

The ordinary [[limit]] and [[colimit]] operations on $D$-diagrams are (as described there) the right and left [[adjoint functor|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 limit|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 equivalence|Quillen equivalent]]).   When these model structures exist (as they do when $C$ is [[combinatorial model category|combinatorial]]), limit and colimit constitute then two $sSet$-enriched [[Quillen adjunction]]s

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

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)-functor]]s (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]]-[[enriched category|enrichment]] one elegant way to do so is in terms of suitable [[weighted limit]]s 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 space]]s or [[spectrum|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 equivalence|weak equivalences]] only, while the local definition is formulated in terms of [[homotopy|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](http://arxiv.org/abs/math/0610194).

#### 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-limit|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 [[generalized the|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 {#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 **[[derivator]]s** 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](ResolvedCoEnds).

### Resolved (co)ends
 {#ResolvedCoEnds}

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 [[copower|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 structure]]s $[D,C]_{Reedy}$ 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 [[weighted limit|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 [[copower|tensoring]] over $V$ in the integran) and where the [[right adjoint]] is formed using the [[power|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](#Gambino10)).

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
 {#BarConstruction}

A general way of obtaining resolutions that compute homotopy (co)limits involves [[bar construction]]s. (...)

## 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)-category|(n,r)]], with the obvious convention that $(n,r)+1 = (n+1,r+1)$ and $\infty+1=\infty$.)  For example:

* [[simplicial set]]s are models for $\infty$-[[infinity-groupoid|groupoids]] ($(\infty,0)$-categories), so [[simplicial model category|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 of categories|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](#Gambino10)) (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](/nlab/show/Ho%28Cat%29#OtherLimits).
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
 {#Examples}

### General weighted colimit formula for homotopy colimits {#WeightedColimitFormula}

Let

* $C$ be a [[combinatorial model category|combinatorial]] [[simplicial model category|simplicial]] [[model category]]

* let $D$ be a [[small category|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 [[copower|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 [[coend]]s over the [[copower|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 bifunctor]]s.

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

+-- {: .num_theorem }
###### 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}
###### Proof

By the fact that the [[coend]] over the [[copower|tensor]] appearing here is a [[Quillen bifunctor]].

This is disucssed for instance in section 4 of ([Gambino 10](#Gambino10)).

=--

#### Examples

##### Homotopy colimits of simplicial diagrams

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

+-- {: .num_prop }
###### 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}
###### 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 [[colimit]]s parameterized by such $D$ are [[pushout]]s. Homotopy colimits over such $D$ are **homotopy pushouts**.

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

+-- {: .num_lemma }
###### Observation

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

=--

Moreover

+-- {: .num_lemma }
###### 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 {#HomotopyPullbacks}

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 [[pullback]]s, 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 {#GroupDeloop}

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
 {#DeloopedSubgroupOverGroup}

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
 {#OverSimplicialDiagrams}

Recall that every [[simplicial set]] is the [[colimit]] of its [[category of elements|elements]] ([this Prop.](simplicial+set#SimplicialSetIsColimitOfItsSimplices)). The following says that this is in fact a homotopy colimit, up to equivalence.

+-- {: .num_prop #SimplicialSetIsHomotopyColimitOverItself}
###### 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}
###### 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 <a href="http://ncatlab.org/nlab/show/Reedy+model+structure#SimplexCategory">Reedy model structure -- simplex category</a> 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.](simplicial+set#SimplicialSetIsColimitOfItsSimplices), the "[[co-Yoneda lemma]]").

=--

+-- {: .num_remark}
###### 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](bisimplicial+set#Diagonal) of the corresponding [[bisimplicial set]].

=--

+-- {: .num_remark}
###### 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 localization of model categories|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]].

\begin{lemma}
The fat simplex is Reedy cofibrant.
\end{lemma}

+-- {: .proof}
###### 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](#Properties) of Reedy model structures, the identity functor $[\Delta, sSet_{Quillen}]_{proj} \to [\Delta, sSet_{Quillen}]_{Reedy}$ is a [[Quillen adjunction|left Quillen functor]], hence preserves cofibrant objects.

=--

+-- {: .num_prop}
###### 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}
###### Proof

The [[coend]] over the [[copower|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 [[Quillen adjunction|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.

=--

+-- {: .num_prop}
###### 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](#Dugger)).

### 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.

+-- {: .num_defn}
###### 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 [[coproduct]]s 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.

=--

+-- {: .num_theorem}
###### 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](BarConstruction).

See for instance ([Dugger, part 1](#Dugger)) for an exposition.

#### Higher homotopy van Kampen theorem

+-- {: .num_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 [[weak homotopy equivalence|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
 {#SequentialHocolims}

#### General
 {#CotowerColimits}

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|descent and codescent]] are naturally conceived as homotopy limits. See also [[infinity-stack]].

### Homotopy (co)limits of simplicial pre(sheaves) {#SimpSheaves}

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 [[site]]s, a [[geometric morphism]] $p : Sh(C) \stackrel{\leftarrow}{\to} Sh(C')$  of [[Grothendieck topos|sheaf topos]]es 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 [[∞-stack]]s 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.

+-- {: .num_theorem }
###### Theorem

Let $p : Sh(C) \to Sh(C')$ be a [[geometric morphism]] of [[Grothendieck topos]]es. 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 presheaf|simplicial sheaf]] categories.

Regarded as a functor between the corresponding [[local model structure on simplicial sheaves|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}
###### Proof

This appears as theorem 1.5 in

* [[Charles Rezk]], _Fibrations and homotopy colimits of simplicial sheaves_ ([arXiv](http://arxiv.org/abs/math/9811038))

=--

### More examples and special cases

* [[filtered homotopy colimit]]

## Related concepts

* [[(∞,1)-limit]] [[lax (∞,1)-colimit]]

* [[mapping cone]], [[mapping telescope]]

## References

### General

Classical references:

* {#BousfieldKan} [[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](https://link.springer.com/book/10.1007/978-3-540-38117-4))

* {#Vogt73} [[Rainer Vogt]], *Homotopy limits and colimits*, Math. Z. __134__ (1973) 11-52 [[eudml:171965](https://eudml.org/doc/171965), [doi:10.1007/BF01219090](https://doi.org/10.1007/BF01219090)]

See

* {#Adams74} [[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](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf))

* [[Emily Riehl]], Chapter 5 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

An introduction is

* {#Dugger} [[Daniel Dugger]], _A primer on homotopy colimits_, 2008 ([pdf](http://math.uoregon.edu/~ddugger/hocolim.pdf))

A general overview via universal properties is in the

* [[Georges Maltsiniotis]] lectures, Sevilla (2008)

  I, localizers, ([pdf](https://webusers.imj-prg.fr/~georges.maltsiniotis/Seville/Lecture_I_Localizers.pdf));

  II, [[model categories]],  ([pdf](https://webusers.imj-prg.fr/~georges.maltsiniotis/Seville/Lecture_II_Model_Categories.pdf));

  III, [[derivator]]s, ([pdf](https://webusers.imj-prg.fr/~georges.maltsiniotis/Seville/Lecture_III_Derivators.pdf)):

  IV, summary on derivators ([pdf](https://webusers.imj-prg.fr/~georges.maltsiniotis/Seville/Summary_on_Derivators.pdf))

In

* Wojciech Chacholski, Jerome Scherer, _Homotopy theory of diagrams_ ([arXiv](http://arxiv.org/abs/math.AT/0110316))

is given a global definition of homotopy (co)limit as [4.1, p. 14](http://arxiv.org/PS_cache/math/pdf/0110/0110316v1.pdf#page=19), 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](http://arxiv.org/PS_cache/math/pdf/0110/0110316v1.pdf#page=20)

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

* {#Gambino10} [[Nicola Gambino]], _Weighted limits in simplicial homotopy theory_, Journal of Pure and Applied Algebra Volume 214, Issue 7, July 2010, Pages 1193–1199 ([pdf](http://www.crm.cat/Publications/08/Pr790.pdf), [publisher](http://www.sciencedirect.com/science/article/pii/S0022404909002412)).

A collection of examples and exercises is in

* {#Isaacson} [[Samuel Isaacson]], _Excercises in homotopy colimits_ ([pdf](http://www-math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf))

See also

* [[Samuel Isaacson]], _A note on unenriched homotopy coends_ ([pdf](http://www.ma.utexas.edu/users/isaacson/PDFs/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](http://www.ams.org/mathscinet-getitem?mr=1214458), [numdam](http://www.numdam.org/item?id=CM_1993__86_2_209_0)

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](http://arxiv.org/abs/math/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](https://arxiv.org/abs/2310.00201).  The case of [[chain complexes]].

* Sergey Arkhipov, Sebastian Ørsted, *Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories*, [arxiv](https://arxiv.org/abs/1807.03266).  A version of [Weighted limits in simplicial homotopy theory](#Gambino10) that doesn't require simplicial enrichment.

* Nicola Gambino, _Homotopy limits for 2-categories_ ([pdf](http://www.lacim.uqam.ca/~gambino/homotopy.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 [[strict 2-limit|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](http://arxiv.org/abs/math.AG/9807049).  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](http://arxiv.org/abs/1104.0646))

* MathOverflow question: [universal-problem-that-motivates-the-definition-of-homotopy-limits](http://mathoverflow.net/questions/38047/what-is-the-universal-problem-that-motivates-the-definition-of-homotopy-limits)

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

* {#HeutsMoerdijk13} [[Gijs Heuts]], [[Ieke Moerdijk]], _Left fibrations and homotopy colimits_ ([arXiv:1308.0704](http://arxiv.org/abs/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](https://github.com/guillaumebrunerie/HoTT/tree/master/Coq/Limits)_

* Jeremy Avigad, [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], _Homotopy limits in Coq_ ([arXiv:1304.0680](http://arxiv.org/abs/1304.0680))

[[!redirects homotopy limits]]
[[!redirects homotopy colimit]]
[[!redirects homotopy colimits]]


Homotopy Limit Functors on Model Categories and Homotopical Categories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

This entry is about the book

* [[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](https://bookstore.ams.org/surv-113-s)

  [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf)

on the theory of [[homotopical categories]] and [[model categories]] -- a presentation of [[(∞,1)-categories]] --, their  [[simplicial localization]]/[[homotopy categories]] and [[derived functors]] such as [[homotopy limit]] functors.

A historically important early manuscript draft of this book is

* [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], [[Model Categories and More General Abstract Homotopy Theory]], [PDF](https://web.math.rochester.edu/people/faculty/doug/otherpapers/dhk.pdf)

The draft serves as an original reference for the [[Kan recognition theorem]] (in §II.8) and the [[Kan transfer theorem]] (in §II.9),
as well as [[cofibrantly generated model categories]] (in Chapter II).
This material did not make it to the book.

Furthermore, this draft originates the modern definition of a [[model category]] (in §I.1.2), modifying the original definition of closed model category by [[Quillen]] by replacing finite (co)limits with small (co)limits and requiring factorizations to be functorial.

## References

Compare also:

* [[Bruno Kahn]], [[Georges Maltsiniotis]], _[[Structures de Dérivabilité]]_

category: reference


A seminal book:

* [[Daniel M. Kan]] and [[Aldridge K. Bousfield]]:

  **Homotopy Limits, Completions and Localizations**

  Lecture Notes in Mathematics 304 (1972; 1987), Springer.

  [doi:10.1007/978-3-540-38117-4](https://link.springer.com/book/10.1007/978-3-540-38117-4)

  ISBN 3-540-06105-3, 0-387-06105-3

introducing [[localization of spaces]] and [[completion of spaces]], as well as [[homotopy limits]] and [[homotopy colimits]].

## Related concepts

* [[localization of spaces]], [[rationalization]]

* [[completion of spaces]]

* [[homotopy limit]]

* [[homotopy colimit]]

category: reference


This page collects links related to

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]],

  \linebreak

  **Homotopy limits, completions and localizations**

  \linebreak

  Lecture Notes in Mathematics, Vol 304, Springer 1972

  [doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4)

  [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bk-yellow.pdf)

on [[p-localization]] and [[p-completion]] (and [[rationalization]]) in [[homotopy theory]].

Precursor:

* {#BousfieldKan71} [[Aldridge Bousfield]], [[Daniel Kan]], *Localization and completion in homotopy theory*, Bull. Amer. Math. Soc. **77** 6 (1971) 1006-1010 [[doi:10.1090/S0002-9904-1971-12837-9](https://doi.org/10.1090/S0002-9904-1971-12837-9), [pdf](https://www.ams.org/journals/bull/1971-77-06/S0002-9904-1971-12837-9/S0002-9904-1971-12837-9.pdf)]

## Related entries

* [[p-localization]]

* [[rationalization]]

* [[totalization]]

(...)

category: reference


homotopy localization at an object > history

see _[[homotopy localization]]_


homotopy n-type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **homotopy $n$-type** is a coarsened variant of the notion of [[homotopy type]], that latter notion being recovered for $n = \infty$.

For instance a [[homotopy 1-type]] has trivial [[homotopy groups]] above degree 2, and a [[homotopy 2-type]] has trivial homotopy groups above degree 3.

Among the most important invariants of a [[topological space]] $X$ or, more generally, of an object $X$ in an [[∞-stack]] [[(∞,1)-topos]] are its [[homotopy group | homotopy groups]] $\pi_k(X)$. We say that an object $X$ for which all $\pi_k(X)$ with $k \gt n$ are trivial is a **homotopy $n$-type**. More precisely, these are the [[n-truncated object of an (infinity,1)-topos|n-truncated objects]] and one says that two object $X$, $Y$ are _of the same homotopy $n$-type_ if there is a zig-zag of morphisms connecting them that induces [[isomorphism | isomorphisms]] on [[homotopy group | homotopy groups]] $\pi_k(X) \stackrel{\simeq}{\to} \pi_k(Y)$ for $0 \leq k \leq n$.

Homotopy $n$-types are, thus, the [[equivalence classes]] of an [[equivalence relation]] imposed on objects in [[Top]] (or objects in another [[(∞,1)-topos]]). Thus, we often say, roughly,  that two spaces 'have the same homotopy $n$-type' if their [[homotopy group | homotopy groups]] agree up to $\pi_n$, and 'a homotopy $n$-type' can equally well be represented by any space having that $n$-type.  This is analogous to the definition of 'a [[real number]]' as an equivalence class of Cauchy sequences.
However, as usual in [[homotopy theory]], merely having isomorphic homotopy groups is not enough; rather there needs to be a map _inducing_ such an isomorphism.  Thus, the relevant equivalence relation relates two spaces when there is a zigzag of maps between them, all inducing isomorphisms on homotopy groups $\pi_k$ for $k\le n$.  One can then show that any space is equivalent, in this sense, to one having trivial homotopy groups above level $n$, so that the other definition is also correct.

The use of [[topological space | topological spaces]] is not, of course, essential; we could just as well use any other structure that models the same [[homotopy theory]], such as [[simplicial set | simplicial sets]], [[simplicial groupoid | simplicial groupoids]], or (for [[connected space | connected spaces]]) [[simplicial group | simplicial groups]].  Moreover, the fact that homotopy $n$-types can be modeled by spaces that are 'homotopically trivial' above level $n$ raises the possibility of finding reasonably complete _algebraic_ models for such $n$-types.

## Definition

A continuous map $X \to Y$ is a **homotopy $n$-equivalence** if it induces isomorphisms on $\pi_i$ for $0 \leq i \leq n$ at each basepoint.  Two spaces share the same **homotopy $n$-type** if they are linked by a zig-zag chain of homotopy $n$-equivalences.

More formally, inverting the $n$-equivalences in [[Top]] gives a [[homotopy category]] $\Ho_n(\Top)$, and two spaces have the same **homotopy $n$-type** if they become isomorphic in $\Ho_n(\Top)$.

For any space $X$, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space $Y$ so that the inclusion of $X$ into $Y$ is a homotopy $n$-equivalence; up to (weak) [[homotopy equivalence]], the result is the same for any space with the same homotopy $n$-type as $X$.  Accordingly, a **homotopy $n$-type** may alternatively be defined as a space with trivial $\pi_i$ for $i \gt n$, or as the unique (weak) [[homotopy type]] of such a space, or as its fundamental $\infty$-[[fundamental infinity-groupoid|groupoid]] (which should be an $n$-[[n-groupoid|groupoid]], by one direction of the [[homotopy hypothesis]]).

One can also construct [[model category|model structures]] on $\Top$ whose homotopy categories are the categories $\Ho_n(\Top)$.  This is one of the original examples of [[Bousfield localization]].  From this perspective, the above replacement of a space by one having trivial $\pi_k$ for $k\gt n$ is an example of _fibrant replacement_.

## Algebraic models

We will use simplicial groups and simplicial groupoids rather than spaces below as they are already partially algebraicised.  So in the definition above, 'space' means a simplicial group(oid) and 'continuous map' means a homomorphism of simplicial group(oid)s.

Considerable effort has gone into finding 'good' algebraic models for (connected) homotopy $n$-types. In low dimensions the results are 'old' or 'classical'.  We will consider connected cases (simplicial groups) only.  The extension to the non-connected case (simplicial groupoids) is 'routine'.

* The 1-type of a connected space is completely determined by its [[fundamental group]], so groups form an algebraic model for [[homotopy 1-type | homotopy 1-types]]. For the non pointed case, we can say [[groupoid | groupoids]] form an algebraic model.

* [[crossed module|Crossed modules]] form an algebraic model for (connected) [[homotopy 2-type | homotopy 2-types]] by a result of Mac Lane and Whitehead,

*  S. Mac Lane and J. H. C. Whitehead, _On the 3-type of a complex_, Proc. Nat. Acad. Sci. U.S.A., 36, (1950), 41 – 48.

The use of crossed modules of groupoids and their [[classifying space]] for the non pointed case is explained under [[homotopy 2-type]].

*  Finding the algebraic model for the $n$-types is just a start.  Ideally one searches for algebraic models of all the higher homotopy structure as well. This was done by [[Jean-Louis Loday]] using the notion of a [[cat-n-group]].

* The method initiated by J.H.C. Whitehead was to approximate homotopy theory by models which analysed particular types of behaviour. One of his most widely followed models is that of [[stable homotopy theory]]. The opposite method was to find algebraic models of restricted classes of spaces, such as 2-types, or with cells in a small range of dimensions.  H.-J. Baues has followed up many of the latter ideas.

It is sensible to regard [[crossed complex|crossed complexes]] as giving a _linear_ model of homotopy types. These crossed complexes are equivalent to [[strict omega-groupoid|strict globular]] $\infty$-groupoids. Although these are restricted model of homotopy types, they are convenient in many aspects, because of the many analogies with the familiar [[chain complex | chain complexes]].

Crossed complexes capture operations of the fundamental groupoid, but not quadratic information such as Whitehead products (for dimensions $\gt 1$). However one can define $n$-fold crossed complexes inductively as crossed complexes internal to $(n-1)$-fold crossed complexes. So one can give the

*(Conjecture) double crossed complexes capture the quadratic information on homotopy types, triple crossed complexes capture the cubic information, etc., etc.

This has the possibility of leading to computations, by applying [[van Kampen theorem | van Kampen theorems]] to specific levels.

## Examples and special cases

[[!include homotopy n-types - table]]

## Related concepts

* [[n-groupoid]]

* [[n-truncated object in an (∞,1)-category]]

* [[model structure for homotopy n-types]]

* [[n-types cover]]

* [[homotopy level]]

## References

### In homotopy theory

Discussion in traditional [[homotopy theory]]:

* {#Spanier66} [[Edwin Spanier]], p. 25 & §8 of:  _Algebraic topology_, McGraw Hill (1966), Springer (1982) [[doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1)]

* [[Hans J. Baues]], *[[Homotopy Types]]* [[pdf](https://archive.mpim-bonn.mpg.de/id/eprint/2171/1/preprint_1994_88.pdf), [[Baues-HomotopyTypes.pdf:file]]], in: [[Ioan Mackenzie James|I. M. James]] (ed.),  *[[Handbook of Algebraic Topology]]*, North Holland (1995) 1-72 [[ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)]

### In homotopy type theory

Discussion of [[homotopy n-types|homotopy $n$-types]]/[[n-truncated object in an (infinity,1)-category|$n$-truncated objects]] in [[homotopy type theory]]:

* [[Univalent Foundations Project]], §7 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* [[Nicolai Kraus]], *Truncation levels in Homotopy Type Theory*, Nottingham (2015) [[pdf](https://eprints.nottingham.ac.uk/28986/1/thesis.pdf), [eprints:28986](https://eprints.nottingham.ac.uk/28986)]

* [[Felix Cherubini]], [[Egbert Rijke]], Thm. 3.10 of: *Modal Descent*, Mathematical Structures in Computer Science , **31** 4 (2021) 363-391  [[doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201), [arXiv:2003.09713](https://arxiv.org/abs/2003.09713)]

[[!redirects n-type]]
[[!redirects n-types]]

[[!redirects homotopy n-types]]
[[!redirects homotopy n-types]]

[[!redirects 1-type]]
[[!redirects 1-types]]
[[!redirects 1-truncated type]]
[[!redirects 1-truncated types]]

[[!redirects 2-type]]
[[!redirects 2-types]]
[[!redirects homotopy 2-type]]
[[!redirects homotopy 2-types]]

[[!redirects 3-type]]
[[!redirects 3-types]]
[[!redirects homotopy 3-type]]
[[!redirects homotopy 3-types]]

[[!redirects truncated type]]
[[!redirects truncated types]]


homotopy n-types - table

| [[homotopy level]] | [[n-truncated object in an (infinity,1)-category|n-truncation]]  |  [[homotopy theory]] | [[higher category theory]] | [[(infinity,1)-topos theory|higher topos theory]] | [[homotopy type theory]] |
|----------------|----------|--------|---------------|--------------------|----------------------|
|  h-level 0     |   (-2)-truncated | [[contractible space]] |  [[(-2)-groupoid]]    |    | [[true]]/[[unit type]]/[[contractible type]] |
|  h-level 1     |   (-1)-truncated | contractible-if-[[inhabited space|inhabited]] | [[(-1)-groupoid]]/[[truth value]] | [[(0,1)-sheaf]]/[[ideal]] |  [[mere proposition]]/[[h-proposition]] |
|  h-level 2     |   0-truncated    | [[homotopy 0-type]] | [[0-groupoid]]/[[set]]  | [[sheaf]]          | [[h-set]] |
|  h-level 3     |   1-truncated    | [[homotopy 1-type]] | [[1-groupoid]]/[[groupoid]] | [[(2,1)-sheaf]]/[[stack]]     | [[h-groupoid]] |
|  h-level 4     |   2-truncated    | [[homotopy 2-type]] | [[2-groupoid]]   |   (3,1)-sheaf/2-stack              | h-2-groupoid |
|  h-level 5     |   3-truncated    | [[homotopy 3-type]] | [[3-groupoid]]   |  (4,1)-sheaf/3-stack       | h-3-groupoid |
|  h-level $n+2$ |   $n$-truncated  | [[homotopy n-type]] | [[n-groupoid]] |[[(n+1,1)-sheaf]]/[[n-stack]] | h-$n$-groupoid |
|  h-level $\infty$ |   untruncated  | [[homotopy type]] | [[∞-groupoid]]  | [[(∞,1)-sheaf]]/[[∞-stack]]                  | h-$\infty$-groupoid |


homotopy of rational maps

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On the [[stable homotopy type|stable]] [[homotopy type]] of [[mapping spaces]] of (regular) [[rational maps]].

Under good conditions, the [[topological subspace|subspace inclusion]] of the space of [[rational maps]] (regular, see Rem. \ref{NotionsOfRationalMaps}),
between given projective [[complex manifolds]] or [[algebraic varieties]],
into the [[mapping space]] of all [[continuous maps]]
induces an [[isomorphism]] in integral [[ordinary homology]] in low degrees, while for maps out of the [[Riemann sphere]] this is even an isomorphism on [[homotopy groups]] in low degrees:

$$
  Maps_{rat}(X_1, X_2)
  \xhookrightarrow{ {homology\; iso} \atop {in\; low\; degree} }
  Maps_{cts}(X_1, X_2)
  \,.
$$

This phenomenon originates in results of [Segal 1979](#Segal79) and is commonly referred to by Segal's name (e.g. "theorems of Segal-type" in [Friedlander & Lawson 1997, Sec. 5.C](#FriedlanderLawson97).

Whenever this holds it provides

1. from left to right: [[homotopy theory|homotopy theoretic]] tools for analyzing [[moduli spaces]] of rational hypersurfaces;

1. from right to left: small algebraic models for [[stable homotopy types|stable]] [[homotopy types]] of [[mapping spaces]]

at least up to some dimension.

## Details

Some remarks on the terminology being used:

\begin{remark}\label{NotionsOfDegree}
**("degree")**\linebreak
  Most or all of the following statement invoke an [[integer]] "degree" of [[continuous functions]]. Beware that this is *not* the [[degree of a continuous function]] (see there) in the usual sense of [[algebraic topology]], except in special cases (such as the archetypical example \ref{TheArchetypicalExample}).
\end{remark}

\begin{remark}\label{NotionsOfRationalMaps}
**("rational maps")** \linebreak
  It it tradition (starting with Segal) to speak of [[rational maps]] in the following, but in the end the focus on *regular rational maps* ("morphisms": e.g. [Friedlander & Lawson 1997, p. 27](#FriedlanderLawson97))), as is necessary to regard them as [[continuous functions]] defined everywhere on the given [[domain]] $X_1$.

In many cases of interest, such as when the [[domain]] $X_1$ is a non-singular [[complex curve]]/[[Riemann surface]] and the [[codomain]] $X_2$ a [[complex projective space]], then all rational maps from $X_1$ to $X_2$ are automatically regular (e.g. [Shafarevich Vol1, Cor. 2.3](rational+map#ShafarevichVol1)).
\end{remark}
For review of more details see [Havlicek 92, §1](#Havlicek92).

### Maps from a Riemann surface to a projective space

* For $n \in \mathbb{N}_+$, consider [[complex projective n-space]]
$\mathbb{C}P^n$.

* Say that a [[continuous map]] $f \;\colon\; \Sigma_2 \to \mathbb{C}P^n$ out of a 2-[[dimension of a manifold|dimensional]] [[manifold]] has *degree* $d \in \mathbb{N}$ (Rem. \ref{NotionsOfDegree}) if the [[pullback in cohomology|pullback]] of the generator $1 \in \mathbb{Z} \simeq H^2\big( \mathbb{C}P^n;\, \mathbb{Z}\big)$ (see [here](complex+projective+space#OrdinaryHomologyAndCohomology)) is

  \[
    \label{DegreeOfAMapToCPn}
    f^\ast(1)
     \,=\,
    d
      \,\in\,
    \mathbb{Z}
      \,\simeq\,
    H^2(\Sigma_2;\, \mathbb{Z})
    \,.
  \]

* For $\Sigma$ a [[compact topological space|compact]] [[connected topological space|connected]] [[Riemann surface]] write $g_\Sigma \in \mathbb{N}$ for its [[genus of a Riemann surface|genus]].

\begin{proposition}\label{SegalOnHomotopyTypeOfRationalMapsIntoCPn}
**(Segal's theorem)** \linebreak
  For $\Sigma$ a [[compact topological space|compact]] [[connected topological space|connected]] [[Riemann surface]], the inclusion
  \[
    \label{RationalMappingSpaceInsideContinuousMappingSpace}
    Maps_{
      {rat}
    }^{deg = d}
    \big(
      \Sigma
      ,\,
      \mathbb{C}P^n
    \big)
    \xhookrightarrow{ \;\; i \;\; }
    Maps_{
      {cts}
    }^{deg = d}
    \big(
      \Sigma
      ,\,
      \mathbb{C}P^n
    \big)
  \]
  of

  * the [[topological subspace]] of [[rational maps]] to [[complex projective n-space]]

    (the *[[moduli spaces]] of rational [[complex curves]] in [[complex projective n-space|$\mathbb{C}P^n$]]*)

  into

  * the [[connected component]] of maps of [[degree of a map|degree]] $d$ of the [[mapping space]] (with the [[compact open topology]]) of all [[continuous maps]]

is

* for $g_\Sigma = 0$ (i.e. $\Sigma$ the [[Riemann sphere]]):

  a [[n-equivalence|$d (2 n -1)$-equivalence]], hence

  an [[isomorphism]] on [[homotopy groups]] in degrees $\lt d (2 n -1)$

  and a [[surjection]] on [[homotopy groups]] in degree $d (2 n  - 1)$

* for $g_\Sigma \geq 1$:

  an [[isomorphism]] on [[ordinary homology|ordinary]] [[homology groups]] in degrees $\lt (d - 2g) (2 n -1)$

  and a [[surjection]] on [[homology groups]] in degree $(d - 2g) (2 n -1)$.

\end{proposition}

This is due to [Segal 1979, Prop. 1.2, 1.3](#Segal79) (bewaring the Note on terminology on p. 44). The analogous statement for rational curves in [[real projective spaces]] is in [Mostovoy 01](#Mostovoy01).

\begin{example}\label{TheArchetypicalExample}
**(the archetypical case)** \linebreak
  In the special case that $n = 1$ and $\Sigma = S^2$ is the [[2-sphere]] with its [[complex structure]], so that both [[domain]] and [[codomain]] are the [[Riemann sphere]] $\mathbb{C}P^1$, Prop. \ref{SegalOnHomotopyTypeOfRationalMapsIntoCPn} says that
$$
    Maps_{
      {rat}
    }^{deg = d}
    \big(
      \mathbb{C}P^1
      ,\,
      \mathbb{C}P^1
    \big)
    \xhookrightarrow{ \;\; i \;\; }
    Maps_{
      {top}
    }^{deg = d}
    \big(
      S^2
      ,\,
      S^2
    \big)
$$

is an isomorphism on [[homotopy groups]] up to degree $\leq d$.
\end{example}
([Segal 1979, Prop. 1.1'](#Segal79))

\begin{remark}\label{RelationToYangMillsMonopoles}
**(relation to [[Yang-Mills monopoles]])** \linebreak
Example \ref{TheArchetypicalExample} controls the classification of [[Yang-Mills monopoles]]. See there for more
\end{remark}

\begin{remark}\label{RelationToGromovWittenTheory}
**(relation to [[Gromov-Witten theory]])**
  A compactification and [[quotient stack]] of the space of rational maps
  in (eq:RationalMappingSpaceInsideContinuousMappingSpace) is considered in [[Gromov-Witten theory]], e.g. [Bertram 2002, p. 9](Gromov-Witten+theory#Bertram02).
\end{remark}

\begin{remark}\label{RelationToTwistorStringTheory}
**(relation to [[twistor string theory]])** \linebreak
  In the context of [[twistor string theory]],
the spaces of rational maps $\Sigma \to \mathbb{C}P^3$ (eq:RationalMappingSpaceInsideContinuousMappingSpace)  are interpreted as [[moduli spaces]] of [[D1-brane]]-[[instantons]] in the [[twistor space]] [[complex projective 3-space|$\mathbb{C}P^3$]] ([Witten 2004, Sec. 3](twistor+string+theory#Witten04)).

Such rational maps are also argued to encode [[scattering amplitudes]] in [[D=4 N=8 supergravity]] ([Cachazo & Skinner 2012](twistor+string+theory#CachazoSkinner12), [Adamo 2015](twistor+string+theory#Adamo15)).

Here the number of poles in the rational function  is the number $n$ of particles in the [[n-point function]], and the genus and degree encode the particle's [[helicity]] and the [[loop order]] of the [[scattering amplitude]].
\end{remark}

\begin{remark}\label{ComaprisonToHomotopicalOkaPrinciple}
**(comparison to the [[homotopical Oka principle]])** \linebreak
  Prop. \ref{SegalOnHomotopyTypeOfRationalMapsIntoCPn} may be compared to the [homotopical Oka principle](Oka+principle#HomotopicalOkaPrinciple), which applies (since $\mathbb{C}P^n$ is an [[Oka manifold]] by [this Prop.](Oka+manifold#ComplexProjectiveSpaceIsOkaManifold)) to the complementary case of connected *non-compact* ("open") Riemann surfaces $\Sigma$ (which are [[Stein manifolds]] by [this Example](Stein+manifold#SteinSurfacesAreOpenRiemannSurfaces)), in which case it says that the corresponding inclusion of the space of [[holomorphic maps]]

$$
    Maps_{
      {hol}
    }
    \big(
      \Sigma
      ,\,
      \mathbb{C}P^n
    \big)
    \xhookrightarrow{ \;\; i \;\; }
    Maps_{
      {top}
    }
    \big(
      \Sigma
      ,\,
      \mathbb{C}P^n
    \big)
$$

induces an isomorphism on all [[homotopy groups]], hence is a [[weak homotopy equivalence]] -- reflecting the fact that non-compactness of the Riemann surfaces and absence of any asymptotic boundary condition provides a large supply of holomorphic functions.
\end{remark}

In fact:

\begin{prop}\label{RationalMapsToCPnAsConfigurationsOfPoints}
  The full [[homotopy type]] of the space of [[pointed topological space|pointed]] rational maps from the [[Riemann sphere]] to [[complex projective space|complex projective $n$-space]] $\mathbb{C}P^n$ of algebraic degree $d$ is that of the [[configuration space of points|configuration space of at most $d$ points]] in $\mathbb{R}^2$ with labels in $S^{2n-1}$:

$$
    Maps_{
      {rat}
    }^{deg = d}
    \big(
      \Sigma
      ,\,
      \mathbb{C}P^n
    \big)
  \;\simeq_{htpy}\;
  \underset{ \leq d}{Conf}
  \big(
    \mathbb{R}^2;
    S^{2k+1}
  \big)
$$
\end{prop}

([Cohen & Shimamoto 91, Theorem 1](#CohenShimamoto91))

### Maps between projective spaces
 {#MapsBetweenProjectiveSpaces}

Generalization to higher dimensional domain spaces:

* Say that the *degree* of a [[rational map]] $f \;\colon\; \mathbb{C}P^{n_1} \to \mathbb{C}P^{n_2}$ between two [[complex projective spaces]] is the degree of the [[polynomials]] that define it.

* In generalization of (eq:DegreeOfAMapToCPn), say that a [[continuous map]] $f \;\colon\; \mathbb{C}P^{n_1} \to \mathbb{C}P^{n_2}$ between two [[complex projective spaces]] has *degree* $d \in \mathbb{N}$ (Rem. \ref{NotionsOfDegree}) if this is the induced factor for [[pullback in cohomology|pullback in]] their second [[integers|integral]] [[ordinary cohomology]] (see [here](complex+projective+space#OrdinaryHomologyAndCohomology))

  \[
    \label{DegreeOfAMapBetweenComplexProjectiveSpaces}
    \array{
      \mathbb{Z}
      \simeq
      H^2\big( \mathbb{C}P^{n_2};\, \mathbb{Z}\big)
      &\xrightarrow{ f^\ast }&
      H^2\big( \mathbb{C}P^{n_1};\, \mathbb{Z}\big)
      \simeq
      \mathbb{Z}
      \\
      1 &\mapsto& d
    }
    \,.
  \]

\begin{prop}\label{HomologyEquivalenceForMapsBetweenComplexProjectiveSpaces}
  For $1 \leq n_1 \leq n_2$, and $d \in \mathbb{N}$, the inclusion
  $$
    Maps^{deg = d}_{rat}
    \big(
      \mathbb{C}P^{n_1}
      ,\,
      \mathbb{C}P^{n_2}
    \big)
    \xhookrightarrow{\;\;\;\;}
    Maps^{deg = d}_{cts}
    \big(
      \mathbb{C}P^{n_1}
      ,\,
      \mathbb{C}P^{n_2}
    \big)
  $$
  of the [[topological subspace]] of [[rational maps]] of algebraic degree $d$ into the [[mapping space]] of [[continuous functions]] of degree $d$ in the sense of (eq:DegreeOfAMapBetweenComplexProjectiveSpaces) induces an [[isomorphism]] on integral [[ordinary homology]] in degrees

  $$
    \leq\,
    \big(
      2 n_1 - 2 n_2 + 1
    \big)
    \big(
      \lfloor(d+2)/2\rfloor + 1
    \big)
    \,,
  $$
  where $\lfloor - \rfloor$ denotes the integer [[floor]] of a rational number.
\end{prop}

([Mostovoy 2006, Theorem 2](#Mostovoy06), with corrected proof in [Mostovoy 2012](#Mostovoy12))

An analogous result for [[real projective spaces]] is in [Adamaszek, Kozlowski & Yamaguchi 2008](#AdamaszekKozlowskiYamaguchi08).

Prop. \ref{HomologyEquivalenceForMapsBetweenComplexProjectiveSpaces} generalizes to the case that the [[codomain]] $\mathbb{C}P^{n_2}$ is allowed to be any [[smooth variety|smooth]] [[toric variety]] ([Mostovoy & Munguia-Villanueva 2012](#MostovoyMunguiaVillanueva12), [Kozlowski & Yamaguchi 2018](#KozlowskiYamaguchi18)).

## Related concepts

* [[Yang-Mills monopole]]

* [[Oka principle]]

* [[Cohomotopy]]

## References

### Maps out of Riemann surfaces

The original theorem for [[rational maps]]/[[continuous maps]] from compact [[Riemann surfaces]] to [[complex projective spaces]]:

* {#Segal79} [[Graeme Segal]], _The topology of spaces of rational functions_, Acta Math. Volume 143 (1979), 39-72 ([euclid:1485890033](https://projecteuclid.org/euclid.acta/1485890033))

Further discussion:

* [[Fred Cohen]], [[Ralph Cohen]], [[B. M. Mann]], [[R. J. Milgram]], _The topology of rational functions and divisors of surfaces_, Acta Math (1991) 166: 163 ([doi:10.1007/BF02398886](https://doi.org/10.1007/BF02398886))

* {#FriedlanderLawson97} [[Eric M. Friedlander]], [[H. Blaine Lawson]], Section 5.C of: *Duality Relating Spaces of Algebraic Cocycles and Cycles*, Topology Volume 36, Issue 2, March 1997, Pages 533-565 ([pdf](https://dornsife.usc.edu/assets/sites/1163/docs/Preprint_versionsPublications/10/projH.pdf))

* [[Ralph L. Cohen]], [[John D. S. Jones]], [[Graeme B. Segal]], *Stability for holomorphic spheres and Morse theory*, in: K. Grove, I. H. Madsen, E. K. Pedersen (eds.) *Geometry and Topology: Aarhus*, Contemporary Mathematics
Volume: 258 (2000)  ([arXiv:math/9904185](https://arxiv.org/abs/math/9904185), [ ISBN:978-0-8218-2158-9](https://bookstore.ams.org/conm-258))

* {#Kamiyama07} Yasuhiko Kamiyama, *Remarks on spaces of real rational functions*, The Rocky Mountain Journal of Mathematics Vol. 37, No. 1 (2007), pp. 247-257 ([jstor:44239357](https://www.jstor.org/stable/44239357))

The analog for rational curves into [[real projective spaces]]:

* {#Mostovoy01} [[Jacob Mostovoy]], *Spaces of Rational Loops on a Real Projective Space*, Transactions of the American Mathematical Society, Vol. 353, No. 5 (May, 2001), pp. 1959-1970 ([jstor:221802](https://www.jstor.org/stable/221802))

Identification of the higher [[homotopy groups]] of $Maps_{rat}(\mathbb{C}P^1, \mathbb{C}P^1)$:

*  [[Martin A. Guest]], [[Andrzej Kozlowski]], M. Murayama, [[Kohhei Yamaguchi]], *The homotopy type of the space of rational functions*, J. Math. Kyoto Univ. 35(4): 631-638 (1995) ([doi:10.1215/kjm/1250518652](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-35/issue-4/The-homotopy-type-of-the-space-of-rational-functions/10.1215/kjm/1250518652.full))

Identification of the full [[homotopy type]] of $Maps_{rat}(\mathbb{C}P^1, \mathbb{C}P^n)$ with a [[configuration space of points]]:

* {#CohenShimamoto91} [[Ralph L. Cohen]], [[Don H. Shimamoto]], *Rational Functions, Labelled Configurations, and Hilbert Schemes*, Journal of the London Mathematical Socienty **43** 2 (1991) 509-528 ([doi:10.1112/jlms/s2-43.3.509](https://doi.org/10.1112/jlms/s2-43.3.509))

Generalization to the case that the [[codomain]] is

... a [[Grassmannian]]:

* [[Frances Kirwan]], *On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles*, Ark. Mat. 24(1-2): 221-275 (1985) ([doi:10.1007/BF02384399](https://www.projecteuclid.org/journals/arkiv-for-matematik/volume-24/issue-1-2/On-spaces-of-maps-from-Riemann-surfaces-to-Grassmannians-and/10.1007/BF02384399.full))

*  [[Benjamin M. Mann]], [[R. James Milgram]],  *Some spaces of holomorphic maps to complex Grassmann manifolds*, J. Differential Geom. 33(2): 301-324 (1991) ([doi:10.4310/jdg/1214446318](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-2/Some-spaces-of-holomorphic-maps-to-complex-Grassmann/10.4310/jdg/1214446318.full))

* {#Havlicek92} John W. Havlicek, *On spaces of holomorphic maps from two copies of the riemann sphere to complex grassmannians*, Stanford University 1992 ([proquest:304009877](https://www.proquest.com/docview/304009877))

... a [[toric variety]]:

* [[Martin A. Guest]], *Configuration spaces and the space of rational curves on a toric variety*, Bull. Amer. Math. Soc. 31 (1994), 191-196 ([doi:10.1090/S0273-0979-1994-00515-4](https://doi.org/10.1090/S0273-0979-1994-00515-4))

* [[Martin A. Guest]], *The topology of the space of rational curves on a toric variety*, Acta Math. 174(1): 119-145 (1995) ([doi:10.1007/BF02392803](https://projecteuclid.org/journals/acta-mathematica/volume-174/issue-1/The-topology-of-the-space-of-rational-curves-on-a/10.1007/BF02392803.full), [arXiv:alg-geom/9301005](https://arxiv.org/abs/alg-geom/9301005))

... a [[flag manifold]]:

* [[C. P. Boyer]], [[B. M. Mann]], [[J. C. Hurtubise]], [[R. J. Milgram]], *The topology of the space of rational maps into generalized flag manifolds*, Acta Mathematica. 1994 Mar 1;173(1):61-101 ([doi:10.1007/BF02392569](https://projecteuclid.org/journals/acta-mathematica/volume-173/issue-1/The-topology-of-the-space-of-rational-maps-into-generalized/10.1007/BF02392569.full))

* [[J. C. Hurtubise]], *Holomorphic maps of a Riemann surface into a flag manifold*, J. Differential Geom. 43(1): 99-118 (1996) ([doi:10.4310/jdg/1214457899](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-43/issue-1/Holomorphic-maps-of-a-Riemann-surface-into-a-flag-manifold/10.4310/jdg/1214457899.full))

Application to the [[quantization]] of [[Skyrmions]] (via their rational map Ansatz, see the references [there](skyrmion#SkyrmionsFromRationalMapsReferences)):

* [[Steffen Krusch]], *Homotopy of rational maps and the quantization of Skyrmions*, Annals of Physics Volume 304, Issue 2, April 2003, Pages 103-127 (doi:10.1016/S0003-4916(03)00014-9)

* [[Steffen Krusch]], *Skyrmions and Rational Maps*, talk at KIAS 2004  ([pdf](http://newton.kias.re.kr/KH04/talks/krusch.pdf), [[Krusch_SkyrmionsAndRationalMaps.pdf:file]])

* [[Steffen Krusch]], *Quantization of Skyrmions* ([arXiv:hep-th/0610176](https://arxiv.org/abs/hep-th/0610176))

### Maps between projective spaces

On maps between [[complex projective spaces]]:

* {#Mostovoy06} [[Jacob Mostovoy]], *Spaces of rational maps and the Stone–Weierstrass theorem*, Topology Volume 45, Issue 2, March 2006, Pages 281-293 ([doi:10.1016/j.top.2005.08.003](https://doi.org/10.1016/j.top.2005.08.003))

corrected proof in:

* {#Mostovoy12} [[Jacob Mostovoy]], *Truncated Simplicial Resolutions and Spaces of Rational Maps*, The Quarterly Journal of Mathematics, Volume 63, Issue 1, March 2012, Pages 181–187 ([doi:10.1093/qmath/haq031](https://doi.org/10.1093/qmath/haq031))

On maps between [[real projective spaces]]:

* {#AdamaszekKozlowskiYamaguchi08} [[Michal Adamaszek]], [[Andrzej Kozlowski]], [[Kohhei Yamaguchi]], *Spaces of algebraic and continuous maps between real algebraic varieties*, Quart. J. Math. 62 (2011), 771–790 ([arXiv:0809.4893](https://arxiv.org/abs/0809.4893), [doi:10.1093/qmath/haq029](https://doi.org/10.1093/qmath/haq029))

On maps from [[real projective space]] to [[complex projective space]]:

* [[Andrzej Kozlowski]], [[Kohhei Yamaguchi]], *Spaces of algebraic maps from real projective spaces into complex projective spaces* ([arXiv:0812.3954](https://arxiv.org/abs/0812.3954)), in: [[Yves Félix]] et al. (eds.) *Homotopy Theory of Function Spaces and Related Topics*, Contemporary Mathematics Volume: 519; 2010 ([ISBN:978-0-8218-4929-3](https://bookstore.ams.org/conm-519))

and [[equivariant homotopy theory|equivariantly]]:

* [[Andrzej Kozlowski]], [[Kohhei Yamaguchi]], *Spaces of equivariant algebraic maps from real projective spaces into complex projective spaces*, RIMS Kôkyûroku Bessatsu B39 (2013), 051−061 ([arXiv:1109.0353](https://arxiv.org/abs/1109.0353), [published pdf](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B39/pdf/B39_006.pdf))

### Maps from projective spaces to toric varieties

On maps from [[complex projective space]] to smooth [[toric varieties]]:

* {#MostovoyMunguiaVillanueva12} [[Jacob Mostovoy]], Erendira Munguia-Villanueva, *Spaces of morphisms from a projective space to a toric variety*, Revista Colombiana de Matematicas **48** 1 (2014) ([arXiv:1210.2795](https://arxiv.org/abs/1210.2795), [published pdf](http://scm.org.co/archivos/revista/Articulos/1129.pdf))

* {#KozlowskiYamaguchi18} [[Andrzej Kozlowski]], [[Kohhei Yamaguchi]], *The homotopy type of spaces of rational curves on a toric variety*, Topology and its Applications Volume 249, 1 November 2018, Pages 19-42 ([doi:10.1016/j.topol.2018.06.006](https://doi.org/10.1016/j.topol.2018.06.006))

[[!redirects homotopies of rational maps]]

[[!redirects homotopy type of spaces of rational maps]]
[[!redirects homotopy types of spaces of rational maps]]

[[!redirects homotopy type of spaces of rational functions]]
[[!redirects homotopy types of spaces of rational functions]]


homotopy product

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#### Homotopy theory
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

__Homotopy products__ are [[Cartesian products]] in [[homotopy theory]], hence are a special case of [[homotopy limits]] for the case that the the indexing [[diagram]] is a [[discrete category]].

## Computation

In any [[model category]], the homotopy product
of a family of objects $\{A_i\}_{i\in I}$
can be computed by [[fibrant resolution|fibrantly replacing]] each $A_i$
and computing the (ordinary) [[Cartesian product]]
of the resulting family $\{\mathrm{R}A_i\}_{i\in I}$ of [[fibrant objects]].

## Examples

\begin{example}
**(homotopy products of simplicial sets)**\linebreak
In [[simplicial sets]] with [[simplicial weak equivalences]] (as in the [[classical model structure on simplicial sets]]), finite homotopy products can be computed by taking their (ordinary) [[Cartesian product]], because [[finite products]] preserve [[simplicial weak equivalences]]. More generally, homotopy products can be computed by applying Kan's [[Ex^∞]] functor to each [[simplicial set]] and taking their (ordinary) [[Cartesian product]].
\end{example}

\begin{example}
**(homotopy products of topological space)**\linebreak
In [[topological spaces]] with [[weak homotopy equivalences]], as in the [[classical model structure on topological spaces]], every object is [[fibrant object|fibant]], so that
homotopy products can be computed as ordinary [[Cartesian products]], which here means [[product spaces]].
\end{example}

\begin{example}
The same applies to the [[relative category]]
of [[chain complexes]] of [[abelian groups]] and [[quasi-isomorphisms]]
([Grothendieck 1957](#Grothendieck1957), section 1.5),

One way to see this is that in the *projective* [[model structure on chain complexes]] (when it exists) again every object is a [[fibrant object]].
\end{example}

## Related notions

* [[Cartesian product]]

* [[homotopy limit]]

* [[homotopy pullback]]

* [[homotopy equalizer]]

* [[homotopy totalization]]

## References

* {#Grothendieck1957} [[Alexander Grothendieck]], _[[Tohoku|Sur quelques points d'algèbre homologique]], Tôhoku Math. J. vol 9, n.2, 3, 1957.

[[!redirects homotopy products]]