category of cubes - exposition

[[!redirects cube category - exposition]]

# Contents
* table of contents
{: toc}

## Idea

The **cube category** $\Box$ encodes one of the main [[geometric shapes for higher structures]]. It is also called the [[cubical category]], although that term can be ambiguous.

Its objects are the standard cellular  "$n$-[[cube|cubes]]", for $n \in \mathbb{N}$ and its morphisms are all possible ways of mapping cubes to each other.

## Definition

+-- {: .un_defn}
###### Definition
The **cube category** is the initial strict [[monoidal category]] $(M, \otimes, I)$ equipped with an object $int$ together with two maps $i_0, i_1: I \to int$ and a map $p: int \to I$ such that $p i_0 = 1_I = p i_1$.
=--

+--{.query}
Do we have a similar definiton of the [[globe category]]?

[[Todd Trimble|Todd]]: None that I know of; the globe category doesn't carry a monoidal structure. But it reminds me that we should create an entry for Joyal's category $\Theta$, used in his definition of weak $\omega$-category, as this cleverly combines globes and simplices.

[[Aleks Kissinger|Aleks]]: What about cubes with connection?
=--

The cube category may also be described as the subcategory of $Set$ whose objects are powers $2^n$ of $2 = \{0, 1\}$, $n \geq 0$, and whose morphisms are generated by degeneracy maps $2^m \to 2^n$ which delete a coordinate and face maps which insert a 0 or 1 without modifying the order of coordinates. The cartesian product on $Set$ restricts to a monoidal product $\otimes$ on this subcategory, giving a strict monoidal category and indeed a [[PRO|pro]]. The basic face maps are the two inclusions $\delta^0, \delta^1: 1 \to 2$, the basic degeneracy is the map $\sigma: 2 \to 1$, and then the general face and degeneracy maps are

$$\delta_i^\varepsilon = 2^{i-1} \otimes \delta^\varepsilon \otimes 2^{n-i}: 2^{n-1} \to 2^n, \qquad \sigma_i = 2^{i-1} \otimes \sigma \otimes 2^{n-i}: 2^n \to 2^{n-1}$$

These satisfy the cubical identities:

> ... to be inserted ...

### Variants

The category of cubes described above has also been described as the _restricted category of cubes_ (see the [paper](http://www.emis.de/journals/TAC/volumes/11/8/11-08.pdf) by Grandis and Mauri). It may be augmented in several directions, at various levels of [[doctrine|doctrinal]] strength, as follows:

> ... to be completed? ...

### Remarks

* The cube category is used to define [[cubical set|cubical sets]].

* The object $int$ may be thought of as the "generic interval" and the monoidal unit $I$ as a point; $x^{\otimes n}$ thus becomes the combinatorial $n$-cube. Indeed, the [[cubical set]] represented by $I$ is the standard cubical 0-cube, while the cubical set represented by $int$ is the standard cubical 1-cube.

* An explicit description of the cube category by [[presentation of a category by generators and relations|generators and relations]] is in section 2 of

  * Sjoerd Crans, _Pasting schemes for the monoidal biclosed structure on $\omega-Cat$_ ([web](http://crans.fol.nl/papers/thten.html), [ps](http://crans.fol.nl/papers/thten.ps.gz), [[thten.pdf|pdf:file]])

* Among all [[geometric shapes for higher structures]] cubes are best suited for describing Gray-like [[tensor product|tensor products]] of higher structures: there is geometrically obvious way in which to combine the $n$-cube $[n]$ and the $m$-cube $[m]$ to the $(n+m)$-cube $[n] \otimes [m] := [n+m]$. This makes $\Box$ into a [[monoidal category]]. It also induces the canonical monoidal structure on [[cubical set|cubical sets]] and then on [[strict omega-category|strict omega-categories]]: the [[Crans-Gray tensor product]].

## Properties

### As a test category

The cube category is a [[test category]]. Hence [[cubical sets]] model [[homotopy types]] (see also _[[model structure on cubical sets]]_). While it is not a [[strict test category]], it can be refined to the category of cubes with "[[connection on a cubical set|cube connection]]", which is. See _[[connection on a cubical set]]_ for more details.

## Related concepts

* [[simplex category]], [[globe category]], [[cell category]]

category: category


category of cycles > history

< [[category of cycles]]


category of descent data

\tableofcontents

## Idea ##

Category of descent data is a particular presentation of a [[2-limit|2-categorical limit]] in [[Cat]], i.e. the [[descent object]], intended to solve the [[descent]] problem. It can be defined in several setups: in a formulation of descent via [[fibered categories]] and [[Grothendieck topologies]], as well as in the (co)[[monadic descent]]. In the first case, the descent category appears as a presentation of the [[descent object]], while in the (co)monadic descent the descent category is given by the [[Eilenberg-Moore category]] of the ([[comonad|co]])[[monad]]. In a particular setup, the two approaches give the same result, see [[Bénabou-Roubaud theorem]].

## Descent along a cover ##

+--{.query}
This kind of descent is described in the article [[stack]] using the bar construction to write a [[sieve]] as a [[colimit]] of representables.
=--

Let $C$ be a site, $p:D\to C$ a fibration, $x\in C_0$, $U_x$ a cover of $x$. $h_x\hookrightarrow C/x$ the full subcategory of arrows, that factor through some $u\rightarrow x$ of the cover; the corresponding functor $C^{op}\rightarrow \Cat$ is then a sieve.
The **category of descent data** (sometimes shortcut to **descent category**) of $p$ along $U_x$ is defined to be the functor category $[h_x,D]$.

## See also ##

* [[descent]]

* [[descent object]]

* [[stack]]

## References ##

* [[Angelo Vistoli]], _Notes on Grothendieck topologies, fibered categories and descent theory_ ([arXiv](http://arxiv.org/abs/math/0412512)) (p.74)

* [[Stacks Project]], ([Tag 02ZC](https://stacks.math.columbia.edu/tag/02ZC))

[[!redirects Descent category]]
[[!redirects descent category]]
[[!redirects the category of descent data]]


category of elements

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _category of elements_ of a [[functor]] $F : \mathcal{C} \to $ [[Set]] is a [[category]] $el(F) \to \mathcal{C}$ sitting over the [[domain]] [[category]] $\mathcal{C}$, such that the [[fiber]] over an [[object]] $c \in \mathcal{C}$ is the set $F(c)$.

This is a special case of the [[Grothendieck construction]], by considering sets as discrete categories.

We may think of [[Set]] as the [[classifying space]] of "[[Set]]-bundles;" see [[generalized universal bundle]].  The category of elements of $F$ is, in this sense, the [[Set]]-bundle classified by $F$.  It comes equipped with a projection to $\mathcal{C}$ which is a [[discrete opfibration]], and provides an equivalence between $\mathbf{Set}$-valued functors and discrete opfibrations.  (There is a dual category of elements that applies to contravariant $\mathbf{Set}$-valued functors, i.e. [[presheaves]], and produces [[discrete fibrations]].)

Forming a category of elements can be thought of as "unpacking" a [[concrete category]]. For example, consider a concrete category $\mathcal{C}$ consisting of two objects $X,Y$ and two non-trivial morphisms $f,g$

The individual elements of $X,Y$ are "unpacked" and become objects of the new category. The "unpacked" morphisms are inherited in the obvious way from morphisms of $\mathcal{C}$.

Note that an "unpacked" category of elements can be "repackaged".

The generalization of the category of elements for functors landing in [[Cat]], rather than just $\mathbf{Set}$, is called the [[Grothendieck construction]].

## Definition

Given a functor $F:\mathcal{C}\to\mathbf{Set}$, the **category of elements** $el(F)$ or $El_F(\mathcal{C})$ (or obvious variations) may be understood in any of these equivalent ways:

* It is the [[category]] whose objects are pairs $(c,x)$ where $c$ is an object in $\mathcal{C}$ and $x$ is an element in $F(c)$ and morphisms $(c,x)\to(c',x')$ are morphisms $u:c\to c'$ such that $F(u)(x) = x'$.

* It is the [[pullback]] along $F$ of the [[generalized universal bundle|universal Set-bundle]] $U : \mathbf{Set}_* \to \mathbf{Set}$

   where $U$ is the [[forgetful functor]] from [[pointed set|pointed sets]] to sets.

* It is the [[comma category]] $(*/F)$, where $*$ is the inclusion of the one-point set $*:*\to \mathbf{Set}$ and $F:\mathcal{C}\to \mathbf{Set}$ is itself:

\begin{center}
\begin{tikzcd}
  \text{El}_F(\mathcal{C})
  \arrow[r, ""]
  \arrow[d, "\pi_F"]
  & \mathbf{Set}_*
  \arrow[d, "U"]
\\
  \mathcal{C}
  \arrow[r, ""]
  & \mathbf{Set}
\end{tikzcd}
\end{center}

* Its [[opposite category|opposite]] is the [[comma category]] $(Y/F)$, where $Y$ is the [[Yoneda embedding]] $\mathcal{C}^{op}\to [\mathcal{C},\mathbf{Set}]$ and $F$ is the functor $*\to [\mathcal{C},\mathbf{Set}]$ which picks out $F$ itself:

\begin{center}
\begin{tikzcd}
  \text{El}_F(\mathcal{C})^{op}
  \arrow[r, "\pi_F^{op}"]
  \arrow[d, ""]
  & \mathcal{C}^{op}
  \arrow[d, "U"]
\\
  *
 \arrow[r, "F"]
  & {[\mathcal{C}, \mathbf{Set}]}
\end{tikzcd}
\end{center}

  $El_F(\mathcal{C})$ is also often written with [[end|coend]] notation as $\int^\mathcal{C} F$, $\int^{c: \mathcal{C}} F(c)$, or $\int^c F(c)$.  This suggests the fact the set of objects of the category of elements is the [[disjoint union]] (sum) of all of the sets $F(c)$.

* It is the (strict) [[oplax colimit]] of the composite $\mathcal{C} \xrightarrow{F} \mathbf{Set} \xrightarrow{disc} \mathbf{Cat}$; see [[Grothendieck construction]].

When $\mathcal{C}$ is a [[concrete category]] and the functor $F:\mathcal{C}\to \mathbf{Set}$ is simply the [[forgetful functor]], we can define a functor

$$Explode(-) := El_F(-).$$

This is intended to illustrate the concept that constructing a category of elements is like "unpacking" or "exploding" a category into its elements.

## Properties

The category of elements defines a functor $el : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat}$.  This is perhaps most obvious when viewing it as an oplax colimit.  Furthermore we have:

+-- {: .num_theorem #ColimitPreserving}
###### Theorem
The functor $el : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat}$ is [[cocontinuous functor|cocontinuous]].
=--
+-- {: .proof}
###### Proof
As remarked above, $el$ is a strict [[weighted colimit|weighted]] [[2-colimit]], hence we have an isomorphism
$$ el(F) \cong \int^{c\in \mathcal{C}} J(c) \times disc(F(c)) $$
where the weight $J:\mathcal{C}^{op} \to \mathbf{Cat}$ is the functor $c\mapsto c/\mathcal{C}$, and $disc:\mathbf{Set}\hookrightarrow \mathbf{Cat}$ is the inclusion of the [[discrete categories]].  But since $disc$ (regarded purely as a 1-functor) has a right adjoint (the functor which sends a -small- category $\mathcal{C}$ into its set of elements $\mathcal{C}_0$), it preserves (1-categorical) colimits.  Since colimits also commute with colimits, the composite operation $\el$ also preserves colimits.
=--
+-- {: .num_theorem #ColimitPreserving2}
###### Theorem
The functor $el\colon \mathbf{Set}^{\mathcal{C}} \to \mathbf{Cat}$ has a right adjoint (which is maybe a more direct way to see that it is cocontinuous).
=--
+-- {: .proof}
###### Proof
By a simple coend computation
$$
\begin{array}{rl}
\mathbf{Cat}(el(F),D)&\cong \mathbf{Cat}\Big(  \int^c J c\times\delta(F  c), D\Big)\\ &\cong \int_c\mathbf{Cat}\big(J c\times \delta(F c),D\big)\\
&\cong \int_c \mathbf{Set}\big(F c,[J c,D]_0\big)\\
&\cong \mathbf{Set}^{\mathcal{C}}(F, K(D))
\end{array}
$$
where $K(D)\colon c\mapsto [J c,D]_0$.
=--

Now for any $\mathcal{C}$, the terminal object of $\mathbf{Set}^\mathcal{C}$ is the functor $\Delta 1$ constant at the [[point]].  The category of elements of $\Delta 1$ is easily seen to be just $\mathcal{C}$ itself, so the unique transformation $F\to \Delta 1$ induces a _projection functor_ $\pi_F: \el(F) \to \mathcal{C}$ defined by $(c,x)\mapsto c$ and $u\mapsto u$.  The projection functor is a [[discrete opfibration]], and can be viewed also as a $\mathcal{C}$-indexed [[family of sets]].  When we regard $\el(F)$ as equipped with $\pi_F$, we have an embedding of $\mathbf{Set}^\mathcal{C}$ into $\mathbf{Cat}/\mathcal{C}$.

Note that the canonical projection $\operatorname{El}(F) \to \mathbf{C}$ is not usually [[full functor|full]]. For example, let $\mathbf{B}\mathbb{N}$ be the one-object category which carries the monoid $(\mathbb{N}, +)$ as its endomorphism monoid, and let $F$ be the action of $(\mathbb{N}, +)$ on the set $\mathbb{N}$ by $n.m = m + n$. Then the image of any hom-set between $k, k'$ is a subsingleton subset of $\mathbb{N}$.

More generally, the [[universal covering groupoid]] of a groupoid is just the category of elements of its action on itself by composition. Since this action is faithful and transitive,  hom-sets in the category of elements are always $0$ or $1$, while objects in the groupoid might have nontrivial automorphism groups.

##Examples

### Representable Presheaves

Let $Y(c):\mathcal{C}^{op}\to \mathbf{Set}$ be a [[representable functor|representable presheaf]] with $Y(c)(d)=Hom_{\mathcal{C}}(d,c)$. Consider the contravariant category of elements $\int_\mathcal{C} Y(c)$ . This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $\mathcal{C}$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in the [[overcategory|slice category]] $\mathcal{C}/c$. Accordingly we see that $\int_\mathcal{C} Y(c)\simeq \mathcal{C}/c$ .

This equivalence comes in handy when one wants to compute [[category of presheaves|slices of presheaf toposes]] over representable presheaves $Y(c)$ since $PSh(\int_\mathcal{C} F) \simeq PSh(\mathcal{C})/F$ in general for presheaves $F:\mathcal{C}^{op}\to \mathbf{Set}$ , whence $PSh(\mathcal{C})/Y(c) \simeq PSh(\mathcal{C}/c)$ . An instructive example of this construction is spelled out in detail at [[hypergraph]].

### Action Groupoid

In the case that $\mathcal{C} = \mathbf{B}G$ is the [[delooping]] [[groupoid]] of a [[group]] $G$, a functor $\varrho : \mathbf{B}G \to \mathbf{Set}$ is a [[permutation representation]] $X$ of $G$ and its category of elements is the corresponding [[action groupoid]] $X/\!/G$.
+-- {: .proof}
###### Proof
This is easily seen in terms of the characterization $el(\varrho)\cong (*/\varrho)$, the category having as objects triples $(*,*; *\to \varrho(*)=X)$, namely elements of the set $X=\varrho(*)$, and as arrows $x\to y$ those $g\in \mathbf{B}G$ such that
\begin{center}
\begin{tikzcd}
  *
  \arrow[r, "x"]
  \arrow[d, "1"]
  & X
  \arrow[d, "g"]
\\
  *
  \arrow[r, "y"]
  & X
\end{tikzcd}
\end{center}
commutes, namely $g . x=\varrho(g)(x)=y$. We can also present the right adjoint to $el(-)$: one must consider the functor $J\colon \mathbf{B}G^{op}\to \mathbf{Cat}$, which represents $G$ in $\mathbf{Cat}$, and sends the unique object $*\in \mathbf{B}G$ to $*/\mathbf{B}G\cong G/\!/G$, the _left_ action groupoid of $G$. The functor $J$ sends $h\in G$ to an automorphism of $G/\!/G$, obtained multiplying _on the right_ $x\to g x$ to $x h\to x g h$.

Now for any category $D$, $K( D)(*)$ is exactly the set of functors $[G/\!/G, D]$, which inherits from $G/\!/G$ an obvious action: given $F\in [G/\!/G,  D]$ we define $F^h=J(h)^*F=F \circ J(h) \colon g \mapsto F(g h)$.
=--
### Category of simplices

For a [[simplicial set]] regarded as a [[presheaf]] on the [[simplex category]], the corresponding category of elements is called its _[[category of simplices]]_. See there for more.

## Related concepts

* [[Grothendieck construction]]

* [[(∞,1)-Grothendieck construction]]

* [[semi-direct product]]

## Reference

A very nice introduction emphasizing the connections to [[monoid|monoid theory]] is ch. 12 of

* [[Michael Barr]], [[Charles Wells]], _Category Theory for Computing Science_ , Prentice Hall 1995³. ([TAC reprints no.22 (2012)](http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html))

[[!redirects Exploding a Category]]


category of factorizations

# Categories of factorizations
* table of contents
{: toc}

## History

In his monograph on (co-)extensions of monoids, Leech introduced a construction that had also been used for other purposes by MacLane, see [[twisted arrow category]].  A short time later Charles Wells distributed a preprint which made the short trip from monoids to small categories with a fixed set of objects.  This introduced the constructions below but then the subject lay fallow until work by [[Baues]] and Wirshing introduced what is now usually called [[Baues-Wirsching cohomology]].  Compare also [[factorization category]].

## Definition

The following constructions were used by Baues and Wirsching and we will more or less adapt their terminology and notation.

Given a small category $I$, one defines a __category of factorizations__ $FI$ as follows. The objects of $FI$ are the morphisms of $I$. A morphism from $x:i\to j$ to $y : k\to l$ is a commutative square of the form (note the direction of arrows!)
$$\array{
i & \stackrel{v}\longleftarrow & k\\
{x}^{\mathllap{}}\downarrow&&\downarrow{y}^{\mathrlap{}}\\
j &\stackrel{u}\longrightarrow & l
}$$
In other words, it is a pair $(u,v)$ which factorizes $y = u\circ x \circ v$. The composition is defined in the obvious way: $(u',v')\circ (u,v) := (u'\circ u, v\circ v')$. For a morphism $f$ in $I$, one usually denotes $D(f)$ by $D_f$ and uses the abbreviation $D_i = D_{id_i}$ for every object $i$ in $I$. Other conventions include
$u_* = D(u,Id) : D_x\to D_{ux}$ and $v^* = D(Id, v) : D_x\to D_{xv}$.

## Natural systems

A __natural system__ on $I$ is a functor $D : FI\to Ab$.

The evident notion of a non-Abelian natural system does not behave that well. A lax version (introduced by Wells) handles that case.

Simple examples of natural systems are provided from the consideration of the sequence of functors

$$
FI \stackrel{(i\overset{x}\to j)\mapsto (i,j)}\longrightarrow
I^{op}\times I \stackrel{(i,j)\mapsto j}\longrightarrow I
$$
which implies that any functor $I\to Ab$ and, more generally, any functor $I^{op}\times I\to Ab$ provides (after precomposing with the canonical functor from $FI$) a natural system on $I$.

Natural systems on $I$ are the coefficients of a __Baues-Wirsching cohomology__ of small categories. Namely, the Baues-Wirsching cochain complex $C^*(I,D)$ has

$$
C^n(I,D) = \prod D_{x_1\cdots x_n}
$$

where the product is over all $i_n\stackrel{x_n}\longrightarrow \ldots \stackrel{x_1}\longrightarrow i_0$.

## References

(A more extensive list may be found at the entry on [[Baues-Wirsching cohomology]].)

* [[H.-J. Baues]], _[[Algebraic Homotopy]]_, Cambridge studies in advanced mathematics  __15__, Camb, Univ. Press 1989.

* [[H.-J. Baues]], G. Wirsching, _Cohomology of small categories_, J. Pure Appl. Alg. __38__ (1985), 187-211.

* T. Pirashvili, _On the center and Baues-Wirsching cohomology_, Georgian Math. J. __16__ (2009) 1, 131-144

* [[C. Wells]], _Extension theories for categories (preliminary report)_ (1979), [available here](
http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf).

[[!redirects category of factorizations]]
[[!redirects category of factorisations]]
[[!redirects categories of factorizations]]
[[!redirects categories of factorisations]]
[[!redirects natural system]]
[[!redirects natural systems]]


category of fibrant objects

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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#

* table of contents
{: toc}

## Idea

A _category of fibrant objects_ is a [[category with weak equivalences]] equipped with extra structure somewhat weaker than that of a [[model category]].

The extra structure of fibrations _and_ cofibrations in a [[model category]] is, while convenient if it exists, not carried by many [[categories with weak equivalences]] which nevertheless admit many constructions in [[homotopy theory]]. Even if they do, sometimes the cofibrations are intractable in practice.

A _category of fibrant objects_ is essentially like a [[model category]] but with all axioms concerning the cofibrations dropped, the concept of fibrations retained ("[[fibration category]]") and assuming that all objects are fibrant (whence the name). It turns out that this is sufficient for many useful constructions. In particular, it is sufficient for giving a convenient construction of the [[homotopy category]] in terms of [[span]]s of length one. This makes categories of fibrant objects useful in [[homotopical cohomology theory]].

## Definition

+-- {: .num_defn}
###### Definition

A **category of fibrant objects** $\mathbf{C}$ is

* a [[category with weak equivalences]], i.e equipped with a subcategory
  $$
    Core(\mathbf{C}) \hookrightarrow W \hookrightarrow C
  $$
  where $f \in Mor(W)$ is called a **weak equivalence**;

* equipped with a further subcategory
  $$
    Core(\mathbf{C}) \hookrightarrow F \hookrightarrow C
    \,,
  $$
  where $f \in Mor(F)$ is called a **fibration**

  Those morphisms which are both weak equivalences and fibrations are called **acyclic fibrations** .

This data has to satisfy the following properties:

* $C$ has [[finite products]], and in particular a [[terminal object]] ${*}$;

* the [[pullback]] of a fibration along an arbitrary morphism exists, and is again a fibration;

* acyclic fibrations are preserved under [[pullback]];

* weak equivalences satisfy [[category with weak equivalences|2-out-of-3]]

* for every object there exists a [[path object]]

  * this means: for every object $B$ there exists at least one object denoted $B^I$ (not necessarily but possibly the [[internal hom]] with an [[interval object]]) that fits into a diagram
  $$
    (B \stackrel{Id \times Id}{\to} B \times B)
    =
    (B \stackrel{\sigma}{\to} B^I \stackrel{d_0 \times d_1}{\to} B \times B)
  $$
  where $\sigma$ is a weak equivalence and $d_0 \times d_1$ is a fibration;

* all objects are _fibrant_, i.e. all morphisms $B \to {*}$ to the terminal object are fibrations.

=--

## Examples

### Full subcategories of model categories
 {#FullSubcategoriesOfModelCategories}

The tautological example is the [[full subcategory]] of any [[model category]] on all objects which are fibrant.

### Right proper model categories
 {#RightProperModelCategories}

Let $M$ be a [[right proper model category]], let $W$ be the class of weak equivalences, and let $F_+$ be the class of morphisms $f$ in $M$ such that any [[pullback]] of $f$ in $M$ is also a [[homotopy pullback]]. Then $M$ together with $W$ and $F_+$ satisfy all the conditions to be a category of fibrant objects _except_ possibly the condition that every morphism $X \to {*}$ in $M$ is in $F_+$; so if we restrict to the [[full subcategory]] of those objects $X$ in $M$ such that $X \to {*}$ is in $F_+$, then we do get a category of fibrant objects.

For example, [[sSet]] via its [[model structure on simplicial sets|standard model structure]] is a category of fibrant objects in this way. The fibrations in this case are not the [[Kan fibrations]] (these also yields a category-of-fibrant-objects structure, via [the above](#FullSubcategoriesOfModelCategories), but a different one) but are the [[sharp maps]].

### $\infty$-Groupoids ##

This includes notably all models for categories of [[infinity-groupoid]]s:

  * the category of [[Kan complex]]es (a full [[subcategory]] of [[SSet]])

  * the category of [[strict omega-groupoid]]s using the [[model structure on strict omega-groupoids]]

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

[[nLab:Kan complex|Kan complexes]] form a [[nLab:category of fibrant objects|Brown category of fibrant objects]].

=--

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

The [[path object]] of any $X$ can be chosen to be the [[nLab:internal hom|internal hom]]

$$
  X^I = [\Delta^1, X]
$$

in with respect to the [[closed monoidal structure on presheaves|closed monoidal structure on]] [[SSet]] with the simplicial 1-[[simplex]] $\Delta^1$.

The morphism $X \to X^I$ is given by the [[simplex category|degeneracy map]] $\sigma_0 : \Delta^0 \to \Delta^1$ as
$$
  X \stackrel{\simeq}{\to}
  [\Delta^0, X]
  \stackrel{[\sigma_0, X]}{\to}
  [\Delta^1, X]
  \,.
$$
This is indeed a weak equivalence, since by the [[simplicial identities]] it is a [[section]] (a [[inverse|right inverse]]) for the morphism
$$
  [\Delta^1, X] \stackrel{[\delta_0,X]}{\to}
  [\Delta^0, X]
  \,.
$$
This map, one checks, has the
[[weak factorization system|right lifting property]]
with respect to all [[boundary of a simplex]]-inclusions
$\partial \Delta^n \to \Delta^n$. By a lemma discussed
at [[Kan fibration]] this means that $[\delta_0,X]$
is an acyclic fibration. Hence $[\sigma_0, X]$, being
its right [[inverse]], is a weak equivalence.

The remaining morphism of the [[path object|path space object]] $X^I \to X \times X$ is
$$
  [\Delta^1, X]
  \stackrel{[\delta_0 \sqcup \delta_1, X]}{\to}
  [\Delta^0 \sqcup \Delta^0, X]
  \stackrel{\simeq}{\to}
  X \times X
  \,.
$$
One checks that this is indeed a [[Kan fibration]].

The stability of fibrations and acyclic fibrations
follows from the above fact that both are characterized
by a [[nLab:weak factorization system|right lifting property]] (as described a [[model structure on simplicial sets]]).

See for instance [section 1](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-1.dvi) of

* Goerss, Jardine, _Simplicial homotopy theory_ .

=--

  Concerning the example of [[Kan complex]]es, notice that [[SSet]] is also a _category of co-fibrant objects_ (i.e. $SSet^{op}$ is a category of fibrant objects) so that [[Kan complex]]es are in fact cofibrant and fibrant. That makes much of the technology discussed below superfluous, since it means that the right notion of $\infty$-morphism between Kan complexes is already the ordinary notion.

  But then, often it is useful to model Kan complexes using the [[Dold-Kan correspondence]], and then the second example becomes relevant, where no longer ever object is cofibrant.

### Simplicial sheaves

The point of the axioms of a category of fibrant objects is that when passing from [[infinity-groupoid]]s to [[infinity-stack]]s, i.e. to [[sheaf|sheaves]] with values in [[infinity-groupoid]]s, the obvious naïve way to lift the model structure from $\infty$-groupoids to sheaves of $\infty$-groupoids fails, as the required lifting axioms will be satisfied only locally (e.g. [[stalk]]wise).

 One can get around this by employing a more sophisticated [[model category]] structure as described at [[model structure on simplicial presheaves]], but often it is useful to use a more lightweight solution and consider sheaves with values in $\infty$-groupoids just as a category of fibrant objects, thereby effectively dispensing with the troublesome lifting property (as all mention of cofibrations is dropped):

+-- {: .num_defn}
###### Definition

**($\infty$-groupoid valued sheaves)**

For $C$ be a [[site]] such that the [[sheaf]] [[topos]]
$Sh(C)$ has [[point of a topos|enough points]],
i.e. so that a morphism $f : A \to B$ in $Sh(X)$
is an isomorphism precisely if its image

$$
  x^* f : x^* A \to x^* B
$$

is a bijection of sets for all [[point of a topos|points]]
([[geometric morphism]]s from $Sh({*}) \simeq Set$)

$$
  x : Set \simeq Sh({*})
   \stackrel{\stackrel{x^*}{\leftarrow}}
     {\stackrel{x_*}{\to}}
  Sh(C)
  \,.
$$

Then let

$$
  \mathbf{C} = SSh(C)
$$

be the full [[subcategory]] of

* [[sheaf|sheaves]] on $C$ with values in the category [[SSet]] of [[simplicial set]]s

* equivalently: [[simplicial object]]s in the [[category of sheaves]] on $C$

on those sheaves $A$
for which each [[stalk]] $x^* A \in SSet$ is a [[Kan complex]].

Define a morphism $f : A \to B$ to be a fibration or a weak equivalence, if on each [[stalk]] $x^* f : x^* A \to x^* B$ is a fibration or weak equivalence, respectively, of Kan complexes (in terms of the standard [[model structure on simplicial sets]]).

=--

**Remarks**

* For instance for $X$ any [[topological space]]
we may take $C = Op(X)$ to be the
[[category of open subsets]] of $X$.
The points of this topos precisely correspond to the
ordinary points of $X$.

  Equipped with its structure as a category of fibrant
objects, simplicial sheaves on $X$ are a model
for [[infinity-stack]]s living **over $X$**
(the way an object $A \in Sh(X)$ is a sheaf "over $X$").

* Or let $C = $ [[Diff]] be a (small model of)
the [[site]] of smooth [[manifold]]s.
The corresponding [[sheaf]] [[topos]],
that of [[smooth space]]s has, up to
isomorphism, one point per natural number, corresponding
to the $n$-dimensional ball $D^n$.

  Equipped with its structure as a category of fibrant
objects, simplicial sheaves on $Diff$ are a model
for [[smooth infinity-stack]]s.

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

$SSh(X)$ with this structure is a category of fibrant objects.

=--

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

The terminal object ${*} = X$ is the sheaf constant on the 0-[[simplex]] $\Delta^0$, which represents the space $X$ itself as a sheaf.

For every simplicial sheaf $A$ and every point $x \in X$ the [[stalk]] of the unique morphism $A \to {*}$ is $x^* A \to x^* {X}$, which is the unique morphism from the [[Kan complex]] $x^* A$ to $\Delta^0$. Since [[Kan complex]]es are fibrant, this is a [[Kan fibration]] for every $x \in X$. So every $A$ is a fibrant object by the above definition.

The fact that fibrations and acyclic fibrations
are preserved under pullback follows from the fact that
the [[stalk]] operation

$$
  x^* : SSh(X) \to SSh(pt) \simeq SSet
$$

is the [[inverse image]] of a [[geometric morphism]]
and hence preserves finite [[limit]]s and in particular [[pullback]]s. So if $f : A \to B$ is a fibration or acyclic fibration in $SSh(X)$ and

$$
  \array{
    A \times_B C &\to& B
    \\
    \downarrow^{\mathrlap{h^* f}} && \downarrow^\mathrlap{f}
    \\
    C &\stackrel{h}{\to}& B
  }
$$

is a [[pullback]] diagram in $SSh(X)$, then for $x \in X$ any point of $X$ also

$$
  \array{
    x^*(A \times_{B} C) &\to& x^*B
    \\
    \downarrow^{\mathrlap{x^* (h^* f)}} && \downarrow^{\mathrlap{x^* f}}
    \\
    x^*C &\stackrel{x^*}{\to}& B
  }
$$

is a [[pullback]] diagram, now of [[Kan complex]]es. Since Kan complexes form a category of fibrant objects,
by the above, it follows that $x^* (h^* f)$ is a fibration or acyclic fibration of Kan complexes, respectively. Since this holds for every $x$, it follows that $h^* f$ is a fibration or acyclic fibration, respectively, in $SSh(X)$.

Recall that a functorial choice of [[path object]] for a [[Kan complex]]e $K$ is the [[internal hom]] $[\Delta^1, K]$ with respect to the [[closed monoidal structure on presheaves|closed monoidal structure on]] [[simplicial set]]s:

$$
  K \stackrel{=}{\to} [\Delta^0, K]
  \stackrel{s_0}{\to}
  [\Delta^1, K]
  \stackrel{d_0 \times d_1}{\to}
  [\Delta^0, K] \times [\Delta^0, K]
  \stackrel{=}{\to}
  K \times K
  \,,
$$

where $s_i$ and $d_i$ denote the [[simplicial set|degeneracy and face maps]], respectively.

For $A \in SSh(X)$ let $[\Delta^1,A]$ denote the sheaf
$$
  [\Delta^1,A] : U \mapsto [\Delta^1,A(U)]
  \,,
$$
where on the left we have new notation and on the right we have the [[internal hom]] in [[SSet]].

(The notation on the left defines the way in which $SSh(X)$ is [[copower]]ered  over [[SSet]]).

We want to claim that $[\Delta^1,A]$ is a [[path object]] for $A$.

To check that $[\Delta^1,A]$ is fibrant, let $x \in X$ be any point and consider the [[stalk]] $x^* [\Delta^1,A] \in SSet$. We compute laboriously

$$
  \begin{aligned}
    x^* [\Delta^1,A] &\simeq colim_{U \ni x} [\Delta^1,A(U)]
    \\
    &\simeq colim_{U \ni x}
     SSet(\Delta^1 \times \Delta^\bullet, A(U))
    \\
    &\simeq
      ([n] \mapsto colim_{U \ni x}
     SSet(\Delta^1 \times \Delta^\n, A(U))
    \\
    &
    \simeq  ([n] \mapsto colim_{U \ni x}
     \int_{[k] \in \Delta}
      Set(\Delta([k],[1])\times\Delta([k],[n]), A(U)_k)
    \\
    & \simeq
      ([n] \mapsto
      \int_{[k \leq n+1] \in \Delta}(
        colim_{U \ni x}
        Set(\Delta([k],[1])\times\Delta([k],[n]), A(U)_k
      )
    \\
    &\simeq
      ([n] \mapsto
      \int_{[k] \in \Delta|_{n+1}}(
       Set(\Delta([k],[1])\times\Delta([k],[n]),
       colim_{U \ni x} A(U)_k
      )
    \\
    &\simeq
      ([n] \mapsto
      \int_{[k] \in \Delta|_{n+1}}(
       Set(\Delta([k],[1])\times\Delta([k],[n]),
       (colim_{U \ni x} A(U))_k
      )
    \\
    &\simeq
     [\Delta^1, colim_{U \ni x} A(U)]
    \\
    & \simeq
     [\Delta^1, x^* A]
)
  \end{aligned}
$$

Where the

* first step is the general formula for the [[stalk]];

* second step is the formula for the [[internal hom]] in the [[closed monoidal structure on presheaves|closed monoidal structure on simplicial sets]];

* third step is the fact that colimits of presheaves are computed objectwise (see examples at [[colimit]]);

* the fourth step is the definition of the
  [[SSet]]-[[enriched functor category]] by an [[end]]

* the fifth step uses that

  * the end truncates to a finite limit with $k \leq n+1$ since $\Delta^1 \times \Delta^n$ is $(n+1)$-[[simplicial skeleton|skeletal]]

  * and that the colimit is over a [[filtered category]]

  * and that [[limits and colimits by example|filtered colimits commute with finite limits]];

* the sixth step uses that the set $\Delta([k],[1])\times \Delta([k],[n])$ is finite, hence a [[compact object]] so that the colimit can be taken into the hom;

* the seventh step uses again that colimits of presheaves are computed objectwise

* the remaining steps then just rewind the first ones, only that now $A(U)$ has been replaced by $colim_{U \ni x} A(U)$.

That the morphism $A \to [\Delta^1,A]$ is a weak equivalence and that $[\Delta^1,A] \stackrel{d_0 \times d_1}{\to} A \times A$ is a fibration follows similarly by taking the [[stalk]] colimit inside to reduce to the statement that
$x^* A \to [\Delta^1,x^* A]$ is a weak equivalence and
$[\Delta^1,x^* A] \stackrel{d_0 \times d_1}{\to} x^*A \times x^* A$ is a fibration, using that $[\Delta^1,x^*A]$ is a [[path object]] for the [[Kan complex]] $x^* A$.

=--

The category of fibrant objects $SSh(X)$
is in fact the motivating example in [[BrownAHT]]. Notice that the [[homotopy category]] in question coincides with that using the [[model structure on simplicial presheaves]], so that the category of fibrant objects of stalk-wise Kan sheaves is a model for the homotopy category of [[infinity-stack]]s.

#### Example

Let $G$ be a topogical [[group]] and recall that
$\mathbf{B} G$ denotes the corresponding one-object
[[groupoid]].

For $X$ a [[topological space]] and
$U$ an open subset, let $C(U, G) \in Set$
be the set of continuous maps from $U$ into
$G$. This set naturally is itself a group, so
that to each $U \subset X$ we may associuate the
one-object groupoid
$$
  U \mapsto \mathbf{B} C(U,G)
  \,.
$$
By postcomposition this with the [[nerve]] operation
we obtain an assignment of [[Kan complex]]es to
open subsets:

$$
  U \mapsto N \mathbf{B} C(U,G)
  \,.
$$

In degree 0 this is the constant [[sheaf]]

$$
  (N \mathbf{B}(-,G))_0 : U \mapsto {*}
$$

while in degree 1 this is the [[sheaf]] of $G$-valued functions

$$
  (N \mathbf{B}(-,G))_1 : U \mapsto C(U,G)
  \,.
$$

When the context is understood, we will just
write $\mathbf{B}G$ again for this $\infty$-groupoid
valued sheaf

$$
  \mathbf{B}G := N \mathbf{B} C(-,G)
  \,.
$$

### Slice categories

Let $\mathbf{C}$ be a category of fibrant objects, with fibrations $F \subset Mor(\mathbf{C})$ and weak equivalences $W \subset Mor(\mathbf{C})$.

For any object $B$ in $\mathbf{C}$, let $\mathbf{C}_B^F$ be the category of fibrations over $B$ (a full subcategory of the [[slice category]] $\mathbf{C}/B$):

* objects are fibrations $A \to B$ in $\mathbf{C}$,

* morphisms are commuting triangles
  $$
    \array{
      A &&\to&& A'
      \\
      & {}_{\in F}\searrow && \swarrow_{\in F}
      \\
      && B
    }
  $$
  in $\mathbf{C}$.

There is an obvious [[forgetful functor]] $\mathbf{C}_B^F \to \mathbf{C}$, which induces notions of weak equivalence and fibration in $\mathbf{C}_B^F$.

+-- {: .num_lemma}
###### Lemma
With this structure, $\mathbf{C}_B^F$ becomes a category of fibrant objects.
=--
+-- {: .proof}
###### Proof

Below is proven the _[[factorization lemma]]_
that holds in any category of fibrant objects.
This implies in particular that every morphism

$$
  \array{
    A &&\stackrel{Id \times Id}{\to}&& A \times_B A
    \\
    & {}_{\in F}\searrow && \swarrow_{\in F}
    \\
    &&
    B
  }
$$

may be factored as

$$
  \array{
    A &\stackrel{\in W}{\to}& A^I &\stackrel{\in F}{\to}&
    A \times_B A
    \\
    & {}_{\in F}\searrow && \swarrow_{\in F}
    \\
    &&
    B
  }
  \,.
$$

This provides the [[path object|path space objects]]
in $\mathbf{C}^F_B$.

=--

## Properties

### Simple consequences of the definition
 {#SimpleConsequences}

Before looking at more sophisticated constructions, we record the following direct consequences of the definition
of a category of fibrant objects.

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

For every two objects $A_1, A_2 \in \mathbf{C}$,
the two projection maps
$$
  p_i : A_1 \times A_2 \stackrel{\in F}{\to} A_i
$$
out of their [[product]] are fibrations.

=--

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

Because by assumption
both morphisms $A_i \to {*}$ are fibrations
and fibrations are preserved under pullback

$$
  \array{
    A_1 \times A_2 &\to& A_2
    \\
    \;{}^{p_1}\downarrow^{\mathrlap{\Rightarrow \in F}} && \downarrow^{\mathrlap{\in F}}
    \\
    A_1 &\to&  {*}
  }
  \,.
$$

=--

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

For every object $B \in \mathbf{C}$ and
everey [[path object]] $B^I$ of $B$,
the two morphisms
$$
  d_i : B^I \stackrel{\in W \cap F}{\to} B
$$
(whose product $d_0 \times d_1$, recall, is required to be
a fibration) are each separately acyclic fibrations.

=--

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

By the above lemma
$d_i : B^I \stackrel{d_0 \times d_1}{\to} B \times B
  \stackrel{p_i}{\to} B$ is the composite of two fibrations
 and hence itself a fibration.

Moreover, from the diagram

$$
  \array{
    B
    &\stackrel{\simeq}{\to}&
    B^I
    &\stackrel{d_0 \times d_1}{\to}&
    B \times B
    \\
    &&&\searrow^{d_i}& \downarrow^{\mathrlap{p_i}}
    \\
    & \searrow^{Id}&&& B
  }
$$

one reads off that the 2-out-of-3
property for weak equivalences implies that $d_i$
is also a weak equivalence.

=--

### Generalized universal bundles and the factorization lemma

A central lemma in the theory of categories of fibrant objects is the following [[factorization lemma]].

+-- {: .num_lemma #FactorizationLemma}
###### Lemma (factorization lemma)

For every morphism $f : C \to D$ in a category
$\mathbf{C}$ of fibrant objects, there is an object
$\mathbf{E}_f B$ such that $f$ factors as

$$
  \array{
    && \mathbf{E}_f B
    \\
    & {}^{\sigma_f \in W}\nearrow && \searrow^{p_f \in F}
    \\
    C &&\stackrel{f}{\to}&& B
  }
$$

with

* $p_f$ a fibration

* $\sigma_f$ a weak equivalence that is a
  [[section]] ( a [[inverse|right inverse]]) of an acyclic fibration:
  $$
    Id_{\mathbf{E}_f B}
    =
    (
     C
      \stackrel{\sigma_f}{\to}
      \mathbf{E}_f B \stackrel{\simeq}{\to}
      C
    )
    \,.
  $$

=--

+-- {: .num_remark}
###### Remarks

This is the analog of one of the _factorization axioms_ in a [[model category]] which says that every map factors as an acyclic cofibration followed by a fibration.

Notice that by
[[category with weak equivalences|2-out-of-3]]
this in particular implies that every weak equivalence
$f : C \stackrel{\in W}{\to} B$
is given by a span of acyclic fibrations.

$$
  \array{
    && \mathbf{E}_f B
    \\
    & {}^{\sigma_f \in W}\nearrow &&
    \searrow^{p_f \in F\cap W}
    \\
    C &&\stackrel{f \in W}{\to}&& B
  }
  \,.
$$

In the context of [[Lie groupoid]] theory these are known as the [[Morita equivalence]]s between groupoids.
There here arise as a special case.
Compare also the notion of [[anafunctor]].

=--

The way the proof of this lemma works, one sees that
this really arises in the wider context of computing
[[homotopy pullback]]s in $C$. Therefore we split the
proof in two steps that are useful in their own right
and will be taken up in the next section on homotopy limits.

+-- {: .num_defn}
###### Definition

For $f : C \to B$ a morphism in $\mathbf{C}$, we say that
the morphism $p_f : \mathbf{E}_f B \to B$
defined as the composite vertical morphism in the
[[pullback]] diagram

$$
  \array{
   \mathbf{E}_f B &\stackrel{\simeq}{\to}& C
   \\
   \downarrow && \downarrow^{\mathrlap{f}}
   \\
   B^I &\stackrel{d_0}{\to}& B
   \\
   \downarrow^{\mathrlap{d_1}}
   \\
   B
  }
$$

for some [[path object|path space object]] $B^I$
is the [[generalized universal bundle]] over $B$
relative to $f$.

=--

The universal bundle terminology is best
understood from the following example

+-- {: .num_example}
###### Example

Consider the category of fibrant objects given by
[[Kan complex]]es or just [[strict omega-groupoid]]s.

For $G$ an ordinary [[group]] write $\mathbf{B} G$
for the corresponding [[groupoid]].
When regarding $G$ as a constant [[simplicial group]]
the corresponding [[Kan complex]] is often denoted
$\bar W G$ (see [[simplicial group]]) but we shall
just write $\mathbf{B} G$ also for this Kan complex,
for simplicity.

The corresponding [[path object]] is given by the [[groupoid]]
(or its corresponding Kan complex)
$$
  (\mathbf{B} G)^I
  =
  [\Delta^1, \mathbf{B} G ]
  =
  G \backslash \backslash G//G
$$
where the right denotes the [[action groupoid]]
of $G \times G$ acting on $G$ by left and right multiplication.

Let ${*} : {*} \to \mathbf{B} G$ be the unique
morphism from the [[point]] into $\mathbf{B} G$.
The corresponding generalized universal bundle is

$$
  \mathbf{E}_{*} G = G//G
$$

the [[action groupoid]] of $G$ acting on  itself
from just the right. (The corresponding [[Kan complex]]
is traditionally denoted $W G$ when thought of as a
[[simplicial group]]).

That $G//G \to \mathbf{B}G$ is indeed the
universal $G$-[[principal bundle]]
(under the [[homotopy hypothesis|Quillen equivalence of]]
[[Kan complex]]es and [[topological space]]s) is an
old result of Segal
(as described at [[generalized universal bundle]]).

=--

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

The morphism $p_f : \mathbf{E}_f B \to B$ is
a fibration.

=--

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

The defining [[pullback]] diagram for $\mathbf{E}_f B$
can be refined to a double pullback diagram as follows

$$
  \array{
    \mathbf{E}_f B
     &\stackrel{\in F}{\to}&
     C \times B
     &\stackrel{p_1}{\to}&
     C
     \\
     \downarrow && \downarrow^{\mathrlap{f \times Id}} && \downarrow^{\mathrlap{f}}
     \\
     B^I &\stackrel{d_0 \times d_1 \in F}{\to}&
     B \times B &\stackrel{p_1}{\to}&
     B
     \\
     \downarrow^{\mathrlap{d_1}} & \swarrow_{p_2 \in F}
     \\
     B
  }
  \,.
$$

Both squares are pullback squares. Since
pullbacks of fibrations are fibrations, the morphism
$\mathbf{E}_f B  \to C \times B$ is a fibration.

By one of the lemmas above, also the projection map
$p_i : B \times B \to B$ is a fibration.

The above diagram exibits $p_f$ as the the composite

$$
  \begin{aligned}
    p_f &: \mathbf{E}_f B \to C \times B
    \stackrel{f \times Id}{\to}
    B \times B
    \stackrel{p_2}{\to}
    B
    \\
    & =
    \mathbf{E}_f B \to C \times B
    \stackrel{p_2}{\to}
    B
  \end{aligned}
$$

of two fibrations. Therefore it is itself a fibration.

=--

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

The morphism $\mathbf{E}_f B \stackrel{\simeq}{\to} C$
has a [[section]] (a [[inverse|right inverse]])
$\sigma_f : C \stackrel{\simeq}{\to}
   \mathbf{E}_f B$ and its composite with
   $p_f$ is $f$:

  $$
    \array{
      \mathbf{E}_f B &\stackrel{\sigma_f}{\leftarrow}&&
      C
      \\
      \downarrow^{\mathrlap{p_f}} && \swarrow_{f}
      \\
      B
    }
  $$

=--

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

The [[section]]
$$
  \sigma_f = Id \times \sigma \circ f
$$
is the morphism induced via the
universal property of the [[pullback]] by the
[[section]] $\sigma : B  \to B^I$ of
$d_0 : B^I \to B$:

$$
  \array{
    C
    &\stackrel{\sigma_f \in W}{\to}&
    \mathbf{E}_f B
    &\stackrel{\in W \cap F}{\to}&
    C
    \\
    \downarrow^{\mathrlap{f}}
    &&
    \downarrow
    &&
    \downarrow^{\mathrlap{f}}
    \\
    B &\stackrel{\sigma}{\to}&
    B^I
    &\stackrel{d_1 \in W \cap F}{\to}&
    B
    \\
    & {}_{Id}\searrow & \downarrow^{\mathrlap{d_0}}
    \\
    && B
  }
  \;\;\;\;
  =
  \;\;\;\;
  \array{
    C
    &\stackrel{Id}{\to}&
    C
    \\
    \downarrow^{\mathrlap{f}}
    &&
    \downarrow^{\mathrlap{f}}
    \\
    B &\stackrel{Id}{\to}&
    B
  }
  \,.
$$

=--

### More sophisticated consequences of the definition

Using the [[factorization lemma]], one obtaines the following
further useful statements about categories of fibrant
objects:

Recall that plain weak equivalences,
if they are not at the same time fibrations,
are not required by the axioms to be preserved by
[[pullback]]. But it follows from the axioms that
weak equivalences are preserved under pullback along fibrations.

This we establish in two lemmas.

+-- {: .num_lemma #BaseChangePreservesFibrationsAndWeakEquivalences}
###### Lemma

Let

$$
 \array{
  A_1 &&\stackrel{f}{\to}&& A_2
  \\
  & {}_{\in F}\searrow && \swarrow_{\in F}
  \\
  && B
 }
$$

be a morphism of fibrations over
some object $B$ in $\mathbf{C}$
and let $u : B' \to B$ be any morphism in
$\mathbf{C}$. Let

$$
 \array{
  u^*A_1 &&\stackrel{u^* f}{\to}&& u^* A_2
  \\
  & {}_{\in F}\searrow && \swarrow_{\in F}
  \\
  && B'
 }
$$

be the corresponding morphism pulled back along $u$.

Then

* if $f \in F$ then also $u^* f \in F$;

* if $f \in W$ then also $u^* f \in W$.

=--

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

For $f \in F$ the statement follows from
the fact that in the diagram

$$
  \array{
    B' \times_B A_1 &\to& A_1
    \\
    \;\;\downarrow^{\mathrlap{u^* f \in F}}
    && \;\;\downarrow^{\mathrlap{f \in F}}
    \\
    B' \times_B A_2 &\to& A_2
    \\
    \;\downarrow^{\mathrlap{\in F}} && \;\downarrow^{\mathrlap{\in F}}
    \\
    B' &\stackrel{u}{\to}& B
  }
$$

all squares (the two inner ones as well as the
outer one) are [[pullback]] squares,
since pullback squares compose under pasting.

The same reasoning applies for $f \in W \cap F$.

To apply this reasoning
to the case where $f \in W$, we first
make use of the factorization lemma
to decompose $f$ as a right inverse to an
acyclic fibration followed by an
acyclic fibration.

$$
  f : A_1 \stackrel{\in W}{\to} \mathbf{E}_f A_2
  \stackrel{\in W \cap F}{\to} A_2
  \,.
$$

(Compare the definition of the category
of fibrant objects
$\mathbf{C}_B^F$ of fibrations over $B$,
discussed in the example section above.)

Using the above this reduces the proof to showing that
the pullback of the top horizontal morphism of

$$
 \array{
  A_1 &&\stackrel{}{\to}&& \mathbf{E}_f A_2
  \\
  & {}_{\in F}\searrow && \swarrow_{\in F}
  \\
  && B
 }
$$

(here the fibration on the right is the composite
of the fibration $\mathbf{E}_f A_2 \to A_2$ with $A_2 \to B$)

along $u$ is a weak equivalence. For that
consider the diagram

$$
  \array{
    B' \times_B A_1  &\to& A_1
    \\
    \downarrow && \downarrow
    \\
    B' \times_B \mathbf{E}_f A_2
    &\to&
    \mathbf{E}_f A_2
    \\
    \;\;\downarrow^{\mathrlap{\in W \cap F}}
    &&
      \;\;\downarrow^{\mathrlap{\in W \cap F}}
    \\
    B' \times_B A_1 &\to& A_1
    \\
    \;\downarrow^{\mathrlap{\in F}} && \;\downarrow^{\mathrlap{\in F}}
    \\
    B' &\to& B
  }
$$

where again all squares are pullback squares.
The top two vertical composite morphisms are
identities. Hence by 2-out-of-3
the morphism
$B' \times_B E_1 \to B' \times_B \mathbf{E}$
is a weak equivalence.

=--

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

The pullback of a weak equivalence along a fibration
is again a weak equivalence.

=--
\begin{proof}
Let be a fibration $p:E\to B$ and let $e:A\to B$ be a weak equivalence. Without
loss of generality we can assume that $e$ is a section of an acyclic fibration
$q:B\to A$ -- otherwise we first decompose with the factorization lemma and then
pull back along the factors individually.

Consider the following diagram where the four tiled squares are pullbacks,
$f=\langle{p,\mathrm{id}}\rangle$, and the bent back rectangle is also a
pullback.
\begin{tikzcd}[sep = 15]
        Y       \ar[rr,"h"']
                \ar[rd,"g"]
                \ar[rddddd, bend right]
&&      E       \ar[rd,"f" description]
                \ar[rrrd,"\mathrm{id}"]
                \ar[rddddd, bend right,"p"]
\\&     E       \ar[dd,"p"]
                \ar[rr, crossing over,"k"' near end]
&&      X       \ar[rr,"l"']
                \ar[dd]
&&      E       \ar[dd,"p"]
\\\\&   B       \ar[rr, crossing over]
                \ar[dd,"q"]
&&      \bullet \ar[rr]
                \ar[dd]
&&      B       \ar[dd,"q"]
\\\\&   A       \ar[rr,"e"]
&&      B       \ar[rr,"q"]
&&      A
\end{tikzcd}
$l$ is a weak equivalence since $q$ is a acyclic fibration, and by 2-out-of-3 we can
conclude that $f$ is a weak equivalence as well. By the preceding lemma \ref{BaseChangePreservesFibrationsAndWeakEquivalences}, so is
$g$. The goal is to show that $h$ is a weak equivalence, and by 2-out-of-3 it's enough
to check that for $k$. This follows again from 2-out-of-3 since $k$ is a section of
$l$ (because the bottom row of the diagram is an identity).
\end{proof}

+-- {: .num_remark}
###### Remarks

[[model category|Model categories]]
that satisfy this property are called
[[right proper model category|right proper model categories]].

Right properness is a crucial assumption
in the closely related work

* Jardine,
  _Cocycle categories_ ([arXiv](http://arxiv.org/abs/math.AT/0605198))
=--

### Homotopy fiber product

Using the existence of [[path object|path space objects]] one can construct specific [[homotopy pullback]]s
called  _homotopy fiber products_ .

+-- {: .num_defn #HomotopyFiberProducts}
###### Definition

A **homotopy fiber product**
or **homotopy pullback** of two morphisms

$$
  A \stackrel{u}{\to} C \stackrel{v}{\leftarrow} B
$$

in a category of fibrant objects
is the object $A \times_C C^I \times_C B$  defined as the
(ordinary) [[nLab:limit|limit]]

$$
  \array{
    A \times_C C^I \times_C B &&&\to & B
    \\
    &&&& \downarrow^v
    \\
    & &C^I & \stackrel{d_0}{\to}& C
    \\
    \downarrow &&
    \downarrow^{\mathrlap{d_1}}
    \\
    A &\stackrel{u}{\to} & C
  }
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

This essentially says that $A \times_C C^I \times_C B$ is the universal object that makes the diagram
$$
  \array{
     A \times_C C^I \times_C B &\to& B
     \\
     \downarrow && \downarrow^{\mathrlap{v}}
     \\
     A &\stackrel{u}{\to}& C
  }
$$
commute up to [[homotopy]]
(see the section on homotopies for more on that).

=--

+-- {: .num_remark}
###### Remark

These homotopy pullbacks present indeed the correct
[[(infinity,1)-limits]], this is the content of
prop. \ref{PullbacksAlongFibrationsAreCorrectHomotopyPullbacks}
below.

=--

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

The projection

$$
 A \times_C C^I \times_C B \to A
$$
out of a homotopy fiber product is a fibration. If $v : B \to C$ is
a weak equivalence, then this is an acyclic fibration.

=--

The same is of course true for the map to $B$ and the
morphism $u : A \to C$, by symmetry of the diagram.

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

One may compute this [[nLab:limit|limit]] in terms of two consecutive
[[nLab:pullback|pullback]]s in two different ways.

On the one hand we have

$$
  \array{
    A \times_C C^I \times_C B &\to& \mathbf{E}_v C &\to
    & B
    \\
    && \downarrow && \downarrow^{\mathrlap{v}}
    \\
    & &C^I & \stackrel{d_0}{\to}& C
    \\
    \downarrow &&
    \downarrow^{\mathrlap{d_1}}
    \\
    A &\stackrel{u}{\to}& C
  }
$$

where both squares are [[pullback]] squares.

By the above lemma on generalized universal bundles,
the map $\mathbf{E}_v C \to C$ is a fibration. The first
claim follows then since fibrations are stable under pullback.

On the other hand we can rewrite the limit diagram also as
$$
  \array{
    A \times_C C^I \times_C B &\to& && B
    \\
    \downarrow && && \downarrow^{\mathrlap{v}}
    \\
    \mathbf{E}_u C & \stackrel{\in W \cap F}{\to} &C^I &
    \stackrel{d_0 \in W \cap F}{\to}& C
    \\
    \downarrow^{\mathrlap{\in W \cap F}}
    &&
    \;\;\downarrow^{\mathrlap{d_1\in W \cap F}}
    \\
    A &\stackrel{u}{\to} & C
  }
$$

where again both inner squares are [[pullback]] squares.

Again by the above statement on generalized universal
bundles, we have that the morphism
$\mathbf{E}_u C \to C$ is a fibration.
By one of the above propositions, weak equivalences
are stable under pullback along fibrations, hence the pullback
$A \times_C C^I \times_C B \to \mathbf{E}_u C$
of $v$ is a weak equivalence. Since also
$\mathbf{E}_u C \to A$ is a weak equivalence
(being the pullback of an acyclic fibration)
the entire morphism
$A \times_C C^I \times_C B  \to A$ is.

=--

### Homotopies
 {#Homotopies}

+-- {: .num_defn}
###### Definition

Two morphism $f,g : A \to B$ in $C(A,B)$
are

* **right [[homotopy|homotopic]]**, denoted $f \simeq g$, precisely if they fit into a diagram
  $$
    \array{
      && B
      \\
      & {}^f\nearrow & \uparrow^{d_0}
      \\
      A &\stackrel{\eta}{\to}&
      B^I
      \\
      & {}_g\searrow & \downarrow^{\mathrlap{d_1}}
      \\
      && B
    }
  $$
  for some [[path object|path space object]] $B^I$;

* **[[homotopy|homotopic]]**, denoted $f \sim g$,
  if they become right homotopic after pulled back
  to a weakly equivalent domain,
  i.e. precisely if they fit into a diagram

$$
  \array{
    &&
    A
    &\stackrel{f}{\to}& B
    \\
    &{}^{w \in W}\nearrow&&& \uparrow^{d_0}
    \\
    \hat A
    &&
    \stackrel{\eta}{\to}
    &&
    B^I
    \\
    &{}_{w\in W}\searrow & &&
    \downarrow^{d_1}
    \\
    &&
    A
    &\stackrel{g}{\to}&
    B
  }
$$

for some object $\hat A$ and
for some [[path object|path space object]] $B^I$ of $I$

=--

+-- {: .num_remark}
###### Remark

So this says that there is a
right [[homotopy]] between the two morphisms after
both are pulled back to a sufficiently good resolution of
their domain.
=--

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

For $A,B \in \mathbf{C}$, right homotopy
is an equivalence relation on the
[[hom-set]] $\mathbf{C}(A,B)$.

=--

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

This follows by "piecing path spaces together":

Let $B^{I_1}$ and $B^{I_2}$ be two path space
objects of $B$. Then the pullback

$$
  \array{
    B^{I_1 \vee I_2} &\to&  B^{I_2}
    \\
    \downarrow && \downarrow^{d_0}
    \\
    B^{I_1} &\stackrel{d_1}{\to}& B
  }
$$

defines a new path object, with structure maps

$$
  B
  \stackrel{\sigma_1 \times \sigma_2}{\to}
  B^{I_1 \vee I_2}
  \stackrel{(d_0 \circ p_1) \times (d_1\circ p_2)}{\to}
  B \times B
  \,.
$$

So given two right homotopies
with respect to $B^{I_1}$ and $B^{i_2}$ we can paste them
next to each other and deduce a homotopy through
$B^{I_1 \vee I_2}$

$$
  \array{
    && B
    \\
    & {}^f\nearrow &  \uparrow^{d_0^1}
    \\
    A &\stackrel{\eta_1}{\to}&
    B^{I_1}
    \\
    & {}_{g}\searrow& \downarrow^{\mathrlap{d_1^1}}
    & \nwarrow
    \\
    && B && B^{I_1 \vee I_2}
    \\
    & {}^{g}\nearrow & \downarrow^{\mathrlap{d_0^2}}
    & \swarrow
    \\
    A &\stackrel{\eta_2}{\to}& B^{I_2}
    \\
    & {}_h\searrow & \downarrow^{\mathrlap{d_1^2}}
    \\
    &&B
  }
$$

=--

We next similarly want to deduce that not only right homotopy
$f \simeq g$ but also true homtopy $f \sim g$
defines an equivalence relation on [[hom-set]]s
$\mathbf{C}(A,B)$. For that we need the following to lemmas.

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

Every diagram

$$
  \array{
    A &\to& E
    \\
    \;\;\downarrow^{\mathrlap{i \in W}}
    &&
    \;\;\downarrow^{\mathrlap{p \in F}}
    \\
    X
    &\to&
    B
  }
$$

may be refined to a diagram

$$
  \array{
    A &\to & X' &\to& E
    \\
    & {}_{i}\searrow &
    \;\;\downarrow^{\mathrlap{t \in W \cap F}}
    &&
    \;\;\downarrow^{\mathrlap{p \in F}}
    \\
    && X
    &\to&
    B
  }
$$

=--

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

Consider the pullback square

$$
  \array{
    A &\to&
    X \times_B E &\to& E
    \\
    &{}_{i \in W}\searrow& \;\; \downarrow^{\mathrlap{\in F}}
    &&
    \;\; \downarrow^{\mathrlap{\in F}}
    \\
    && X &\to& B
  }
$$

and apply the [[factorization lemma]], lemma \ref{FactorizationLemma}, to factor
the universal morphism $A \to X \times_B E \to E$
into the pullback as

$$
  A \stackrel{\in W}{\to} \mathbf{E} E \stackrel{\in F}{\to}
  E
$$

to obtain the diagram

$$
  \array{
    A &\stackrel{\simeq}{\to}&
    \mathbf{E} E &\to& E
    \\
    &{}_{i \in W}\searrow&
    \;\; \downarrow^{\mathrlap{\in F}}
    &&
    \;\; \downarrow^{\mathrlap{\in F}}
    \\
    && X &\to& B
  }
  \,,
$$

where the middle vertical morphism is still a
fibration, being the composite of two fibrations.
By [[category with weak equivalences|2-out-of-3]]
it follows that it is also a weak equivalence.

=--

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

For $ u : B \to C$ a morphism and $B^I$, $C^I$ choices of path objects,
there is always another path object $B^{I'}$
with an acyclic fibration
$B^I \stackrel{\in W \cap F}{\leftarrow} B^{I'}$
and a span of morphisms of path space objects

$$
  \array{
    B &\stackrel{=}{\leftarrow}& B &\stackrel{u}{\to}& C
    \\
    \downarrow^{\mathrlap{\sigma}} && \downarrow^{\sigma'} && \downarrow^{\mathrlap{\sigma_C}}
    \\
    B^I &\stackrel{\in W \cap F}{\leftarrow}& B^{I'} &\stackrel{\bar u}{\to}& C^I
    \\
    \;\;\downarrow^{d_0 \times d_1}
    && \;\;\downarrow^{\mathrlap{d'_0 \times d'_1}}
    && \;\;\downarrow^{\mathrlap{d_0^C \times d_1^C}}
    \\
    B \times B
    &\stackrel{=}{\leftarrow}&
    B \times B
    &\stackrel{u \times u}{\to}&
    C \times C
  }
$$

=--

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

Apply the lemma above to the square

$$
  \array{
    B &\stackrel{u}{\to}& C
     &\stackrel{\sigma_C}{\to}&
     C^I
    \\
    \downarrow^{\mathrlap{\sigma}} &&&& \downarrow^{\mathrlap{d_0 \times d_1}}
    \\
    B^I &\stackrel{d_0 \times d_1}{\to}& B \times B
    &\stackrel{u \times u}{\to}& C \times C
  }
  \,.
$$

=--

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

Right homotopy $f \simeq g$ between morphisms
is preserved under pre- and postcomposition with a given morphism.

More precisely,
let $f, g : B \to C$ be two homotopic morphisms. Then

* for all morphisms $A \to B$ and $C \to D$ the composites
$A \to B \stackrel{f}{\to} C \to D$ and $A \to B \stackrel{g}{\to} C \to D$
are still right homotopic.

* moreover, the right homotopy may be realized
  with every given choice of
  [[nLab:path object|path space object]] $D^I$ for $D$.

=--

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

We decompose this into two statements:

1. for any $A \to B$ the morphisms $A \to B \stackrel{f,g}{\to} B$ are right homotopic.

1. for any $u : C \to D$ and choice $D^I$ of path object there is an acyclic fibration
   $
     B' \to B
   $
   such that
   $B' \to B \stackrel{f}{\to} C \to D$
   is right homotopic to
   $B' \to B \stackrel{g}{\to} C \to D$
   by a right homotopy $\eta : B' \to D^I$.

The first of these follows trivially.

The second one follows by using the weak functoriality property of path objects
from above: let $B' := B \times_{C^I} C^{I'}$
be the [[pullback]]
in the following diagram

$$
  \array{
    B'
    &\to&
    C^{I'}
    &\stackrel{\bar u}{\to}&
    D^I
    \\
    \;\;\;\downarrow^{\mathrlap{\in W \cap F}}
    &&
    \;\;\;\downarrow^{\mathrlap{\in W \cap F}}
    &&
    \downarrow
    \\
    B
    &\stackrel{\eta}{\to}&
    C^I
    \\
    &{}_{f \times g}\searrow
    & \downarrow
    &&
    \downarrow
    \\
    &&
    C \times C
    &\stackrel{u \times u}{\to}&
    D \times D
  }
$$

=--

We need one more intermediate result
for seeing that homotopy is an equivalence relation

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

* Every diagram

  $$
  \array{
    && B
    \\
    && \downarrow^{\mathrlap{w \in W}}
    \\
    A &\to& C
  }
  $$

  in $\mathbf{C}$ extends to a
  (right) homtopy-commutative diagram
  $$
  \array{
    A' &\to & B
    \\
    \downarrow^{\mathrlap{w' \in W}} && \downarrow^{\mathrlap{w \in W}}
    \\
    A &\to& C
  }
  \,.
  $$

* For every pair of morphisms
  $$
    f, g, A \stackrel{\to}{\to} B
  $$
  and weak equivalence
  $t : B \stackrel{\in W}{\to} C$
  such that there is a right homotopy
  $t \circ f \simeq t \circ g$, there exists
  a weak equivalence $t' : A' \to A$
  such that $f \circ t' \simeq g \circ t'$.

=--

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

* The first point we accomplish this by letting $A' := A \times_C C^I \times_C B$ be the homotopy fiber product
in $C$ of a representative of the pullback diagram.
The lemma
about morphisms out of the
homotopy fiber product says
that $A' \to A$ is a weak equivalence.

* The second point is more work.
  Let $\eta : A \to C^I$ the right homotopy in question.
  We start by considering
  the homotopy fiber product
  $$
    \array{
      D := B \times_C C^I \times_C B
      &\to&&\stackrel{\in W}{\to}& B
      \\
      \downarrow^{\mathrlap{\in W}} &&&&
      \downarrow^{\mathrlap{t \in W}}
      \\
      && C^I &\stackrel{d_0}{\to}&
      C
      \\
      \downarrow && \downarrow^{d_1}
      \\
      B &\stackrel{t \in W}{\to}& C
    }
    \,,
  $$
  where the long morphisms are weak equivalences
  by the lemma on morphisms out of homotopy fiber products.

 Then consider the two universal morphisms
  $$
   (f,\eta,g) : A \to B \times_C C^I \times_C B
  $$
  and
  $$
    (Id, \sigma \circ t, Id) :
    B \stackrel{\in W}{\to} B \times_C C^I \times_C B
  $$
  into that. It follows by 2-out-of-3 that the latter
  is a weak equivalence. Factoring this
  using the factorization lemma produces hence
  $$
    B \stackrel{\in W}{\to} D' \stackrel{\in W \cap F}{\to}
    D
    \,.
  $$

  We know moreover that the product map
  $D \stackrel{\in F}{\to} B \times B$
  is a fibration, as we can rewrite the homotopy
  limit as the pullback
  $$
    \array{
       D &\to& C^I
       \\
       \downarrow && \downarrow^{\mathrlap{\in F}}
       \\
       B \times B &\stackrel{f \times g}{\to}&
       C \times C
    }
    \,.
  $$

  It follows that the composite $D' \to D \to B \times B$
  is a fibration and hence $D'$ a path space object for
  $B$.

  Finally, by setting $A' = A \times_D D'$
  we obtaine the desired right homotopy
  $f \circ t' \simeq g \circ t'$.
  $$
    \array{
      A' &\to& D'
      \\
      \downarrow^{\mathrlap{t'}} && \downarrow
      \\
      A &\to & D &\to & C^I
      \\
      & {}_{f \times g}\searrow & \downarrow
      && \downarrow
      \\
      && B \times B
      &\stackrel{t \times t}{\to}&
      C \times C
    }
    \,.
  $$

=--

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

The relation "$f, g \in C(A,B)$ are homotopic",
$f \sim g$,
is an [[equivalence relation]] on $C(A,B)$.

=--

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

The nontrivial part is to show transitivity.
This now follows with the above lemma
about homtopy commutative composition of
pullback diagrams and then using the
"piecing together of path objects"
used above to show that right homotopy
is an equivalence relation.

=--

+-- {: .num_defn}
###### Definition

For $C$ a category of fibrant objects the category
$\pi C$ is defined to be the category

* with the same objects as $C$;

* with [[hom-set]]s the set of equivalence classes
  $$
   \pi C(A,B) := C(A,B)/_\sim
  $$
  under the above equivalence relation.

* Composition in $\pi C$ is given by composition of
  representatives in $C$.

=--

+-- {: .num_defn}
###### Definition

The obvious functor

$$
  C \to \pi C
$$

is the identity on objects and the projection to
[[equivalence relation|equivalence classes]] on
[[hom-set]].

Let $\pi W \subset Mor(\pi C)$ be the image of the
weak equivalences of $C$ in $\pi C$ under this functor,
and $\pi F$ the image of the fibrations.

=--

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

The weak equivalences in $\pi C$ form  a left multiplicative system.

=--

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

This is now a direct consequence of the above
lemma on homotopy-commutative completions of diagrams.

=--

### The homotopy category
 {#HomotopyCategory}

We discuss now
that the structure of a category of fibrant objects on a
[[homotopical category]] $C$ induces

* a related category $\pi C$

* with a morphism $C \to \pi C$

  * that is the identity on objects,

  * and induces on $\pi C$ a notion of weak equivalences
    $$
      \pi W \subset Mor(\pi C)
    $$
    and fibrations
    $$
      \pi F \subset Mor(\pi C)
    $$

* such that

  * the weak equivalences in $\pi C$ form a [[calculus of fractions|left multiplicative system]] .

This implies the following convenient construction
of the [[homotopy category]] of $C$:

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

For $C$ a category of fibrant objects,
its [[homotopy category]] is
([[equivalence of categories|equivalent]] to) the category
$Ho_C$ with

* the same objects as $C$;

* the [[hom-set]] $Ho_C(A,B)$ for all $A, B \in Obj(C)$
  given naturally by
  $$
    \begin{aligned}
      Ho_C(A,B)
      & \simeq
      colim_{\hat A \stackrel{w\in \pi W}{\to} A}
      \pi C (\hat A,B)
      \\
      & =
      colim_{\hat A \stackrel{f\in \pi W\cap F}{\to} A}
      \pi C (\hat A,B)
    \end{aligned}
    \,.
  $$

=--

Here the colimit is, as described at [[calculus of fractions|multiplicative system]], over the [[opposite category]] of the category $\pi W_A$ or $(\pi F\cap \pi W)_A$
whose objects are weak equivalences $\hat A \stackrel{w \in \pi W}{\to} A$ or acyclic fibrations
$\hat A \stackrel{f \in \pi W\cap F}{\to} A$ in $\pi C$, and whose morphisms are commuting triangles

$$
  \array{
    \hat A &&\stackrel{h}{\to}&& \hat A'
    \\
    & \searrow && \swarrow
    \\
    &&
    A
  }
$$
in $\pi C$ (i.e. for arbitrary $h$).

So more in detail the above colimit is over the functor

$$
  \pi C(-, B)_A :
  (\pi W_A)^{op} \to (\pi C)^{op}
  \stackrel{\pi C(-, B)}{\to}
  Set
  \,,
$$

where the first functor is the obvious forgetful functor.

+-- {: .num_remark}
###### Remark

It is again the factorization lemma above (and using 2-out-of-3 that implies that inverting just the acyclic fibrations in $C$ is already equivalent to inverting all weak equivalences. This means that the above theorem remains valid if the weak equivalences $t : A' \to A$ are replaced by _acyclic fibrations_:

every cocycle
$\array{
   Y &\stackrel{g}{\to}& A
   \\
   {}^\simeq \downarrow^{\mathrlap{f}}
   \\
   X
 }$

out of a weak equivalence is refines by a cocycle out of an acyclic fibrantion, namely

$$
\array{
   \mathbf{E}_f X &\stackrel{\simeq}{\to}&
      Y &\stackrel{g}{\to}& A
   \\
   &{}_{\in F \cap W}\searrow& {}^\simeq \downarrow^{\mathrlap{f}}
   \\
   && X
 }
  \,.
$$

Using acyclic fibrations has the advantage that these are preserved under [[pullback]]. This allows to consistently compose spans whose left leg is an acyclic fibration by [[pullback]]. See also the discussion at [[anafunctor]].

A discussion of this point of using weak equivalences versus acyclic fibrations in the construction of the homotopy category is also in Jardine: [Cocycle categories](http://www.math.uiuc.edu/K-theory/0782/).
=--

We now provide the missing definitions and then the proof
of this theorem.

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

The homotopy categories of $C$ and $\pi C$ coincide:

$$
  Ho_C \simeq Ho_{\pi C}
  \,.
$$

=--

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

By one of the lemmas above, the morphisms
$d_i : B^I \to B$ are weak equivalences
and become [[isomorphism]]s in $Ho_C$.
The [[section]] $\sigma : B \to B^I$
then becomes an [[inverse]] for both
of them, hence the images of $d_0$ and
$d_1$ in $Ho_C$ coincide. Therefore
the above diagram says that homotopic morphisms
in $C$ become equal in $Ho_C$.

But this means that the localization morphism
$$
  Q_C : C \to Ho_C
$$
factors through $\pi C$ as

$$
  Q_C : C \to \pi C \stackrel{Q_{\pi C}}{\to} Ho_C
$$

where $Q_{\pi C}$ sends weak equivalences in $\pi C$
to isomorphisms in $Ho_C$.

The universal property of $Q$ then implies the universal
property for $Q_{\pi C}$

$$
  \array{
    C &\to& \pi C &\to & A
    \\
    \downarrow^{\mathrlap{Q_C}} & \swarrow^{Q_{\pi C}} && \swarrow
    \\
    Ho_C
  }
  \,.
$$

=--

The above theorem on the description of $Ho_C$
now follows from the general formula for
[[localization]] at a
[[calculus of fractions|left multiplicative system]]
of weak equivalences.

### Pointed category of fibrant objects

If the category $C$ of fibrant objects has an initial object which _coincides_ with the terminal object $e$, i.e. a [[zero object]], then $C$ is a [[pointed category]]. In this case we have the following additional concepts and structures.

### Fibers

 For $p : Y \to X$ a fibration, the pullback $F$ in

$$
  \array{
     F &\stackrel{i}{\to}& Y
     \\
     \downarrow && \downarrow
     \\
     e &\to& X
  }
$$

is the **fibre** of $p$ and $i$ is the _fibre inclusion_.
(This is the _kernel_ of the morphism $f$ of [[pointed object]]s)

### Fibration Sequences

(See also [[fibration sequence]])

For $B$ any object and $B^I$ any of its [[path object]]s, the fiber of $B^I \stackrel{d_0 \times d_1}{\to} B \times B$ is the **[[loop space object|loop object]]** $\Omega^{(I)} B$ of $B$ with respect to the chosen path object. This construction becomes independent up to canonical isomorphism of the chosen path space after mapping to the [[homotopy category]] and hence there is a functor

$$
  \Omega : Ho_C \to Ho_C
$$

which sends any object $B$ of $C$ to its canonical loop object $\Omega B$.

Any loop object $\Omega B$ becomes a group object in $Ho_C$, i.e. a group [[internalization|internal to]] $Ho_C$ in a natural way.

### Derived hom-spaces
 {#DerivedHomSpaces}

There is an explicit simplicial construction of the [[derived hom spaces]] for a homotopical category that is equipped with the structure of a category of fibrant objects.
This is described in ([Cisinksi 10](#Cisinski10)) and ([Nikolaus-Schreiber-Stevenson 12, section 3.6.2](#NSS12)).

+-- {: .num_defn #CocycleCategories}
###### Definition

For $\mathcal{C}$ a category of fibrant objects, write for any $X, A \in Obj(\mathcal{C})$

$$
  Cocycle(X,A) , wCocycle(X,A) \in Cat
$$

for the [[categories]] ("categories of [[cocycles]] on $X$ with [[coefficients]] in $A$") whose [[objects]] are [[correspondences]]

$$
  X \stackrel{\simeq}{\longleftarrow}
  \hat X
  \stackrel{}{\longrightarrow}
  A
$$

with the left leg an acyclic fibration (for $Cocycle(X,A)$) or just a weak equivalence (for $wCocycle(X,A)$); and whose morphisms are morphisms of spans

$$
  \array{
    && \hat X
    \\
    & \swarrow && \searrow
    \\
    X && \downarrow && A
    \\
    & \nwarrow && \nearrow
    \\
    && \hat X'
  }
$$

=--

+-- {: .num_prop #CocycleCategoriesPresentDerivedHomSpaces}
###### Proposition

Write $L^H_{we} \mathcal{C}$ for the [[simplicial localization]] of the category of fibrant objects $\mathcal{C}$ at its weak equivalences (hence essentially the [[(infinity,1)-category]] that it presents). Then for all objects $X,A \in Obj(\mathcal{C})$ the canonical maps

$$
  N Cocycle(X,A)
  \to
  N wCocycle(X,A)
  \to
  L^H_{we} \mathcal{C}(X,A)
$$

of [[simplicial sets]] (on the left the [[nerves]] of the cocycle categories of def. \ref{CocycleCategories}, on the right the [[derived hom space]] given by the [[simplicial localization]]) are [[weak homotopy equivalences]].

=--

In other words, $N Cocycle(X,A) \simeq_{whe} N wCocycle(X,A)$ is a model for the correct [[derived hom space]].

From this it follows for instance that

+-- {: .num_prop #PullbacksAlongFibrationsAreCorrectHomotopyPullbacks}
###### Proposition

The [[homotopy fiber products]] in $\mathcal{C}$ as defined in def. \ref{HomotopyFiberProducts} present indeed the correct [[(infinity,1)-limits]].

=--

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

Observe that for each object $X$ the 2-functor $N Cocycle(X,-) \colon \mathcal{C} \to sSet$ of def. \ref{CocycleCategories} sends fibrations
to [[Kan fibrations]] of simplicial sets (the [[horn]]-filling condition comes down to factoring maps through the given fibration, which is possible by pullback along the fibration). Moreover, it is evident that $N Cocycle(X,-)$ preserves ordinary [[pullbacks]]. This means that $N Cocycle(X,-)$ takes pullbacks along a fibration in $\mathcal{C}$ to pullbacks in [[sSet]] one of whose maps is a [[Kan fibration]]. Since the standard [[model structure on simplicial sets]] $sSet_{Quillen}$ is a [[right proper model category]], this means that these are [[homotopy pullbacks]] (as discussed there) in $sSet_{Quillen}$. Finally by prop. \ref{CocycleCategoriesPresentDerivedHomSpaces} this means that the [[derived hom-space]] functor $\mathbb{R}Hom(X,-)$ sends pullbacks along fibrations to [[homotopy pullbacks]] of the correct derived hom-spaces. This means (as discussed for instance at _[[homotopy Kan extension]]_) that the original pullbacks in $\mathcal{C}$ are the correct homotopy pullbacks.

=--

## Application in cohomology theory

When the catgegory of fibrant objects is that of locally Kan simplicial sheaves, the [[hom-set]]s of its
[[homotopy category]] compute generalized notions of [[cohomology]].

At [[abelian sheaf cohomology]] is a detailed discussion of how the ordinary notion of sheaf cohomology arises as a special case of that.

## Related concept

* [[category of cofibrant objects]]

* [[Waldhausen category]]

* [[model category]]

## References

The notion of _category of fibrant objects_ was introduced and the above results obtained in

* [[Kenneth Brown]], _[[BrownAbstractHomotopyTheory.pdf:file]]_, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 ([[BrownAHT]]).

for application to [[homotopical cohomology theory]].

A review is in [section I.9](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-1.dvi) of

* [[Paul Goerss]] and [[Rick Jardine]], 1999, _[[Simplicial homotopy theory]]_, number 174 in Progress in Mathematics, Birkhauser. ([ps](http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html))

There is a description and discussion of this theory and its dual (using cofibrant objects) in

* [[K. H. Kamps]] and [[Tim Porter]], _Abstract Homotopy and Simple Homotopy Theory_ ([GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))

Discussion of embeddings of categories of fibrant objects into model categories is in

* {#Cisinski10} [[Denis-Charles Cisinski]], _Invariance de la K-Théorie
par équivalences dérivées_ ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/eqderkth.pdf))

Also discussion of the [[derived hom spaces]] in categories of fibrant objects is in that article, as well as in section 6.3.2 of

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_ ([arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

and also in

* [[Geoffroy Horel]], *Brown categories and bicategories*, [arxiv](https://arxiv.org/abs/1506.02851)

Usage of categories of fibrant objects for the [[homotopical structure on C*-algebras]] is in :

* [[Otgonbayar Uuye]], _Homotopy theory for $C^\ast$-algebras_,  Journal of Noncommutative Geometry, ([arxiv:1011.2926](http://arxiv.org/abs/1011.2926))

Categories of fibrant objects form a convenient setting for the study of [[homotopy type theory]]:

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

It is shown in the following paper that categories of fibrant objects are themselves fibrant in the [[model structure on categories with weak equivalences]]:

* Lennart Meier, *Fibration Categories are Fibrant Relative Categories*, [arxiv](https://arxiv.org/abs/1503.02036)
 {#Meier}

[[!redirects categories of fibrant objects]]


category of filters

[[!redirects category of filters on sets]]

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

# The category of filters
* table of contents
{: toc}

## Definition

_A [[filter]] $\mathfrak{F}$ on a set $X$_ is
 a set of of subsets of X which has the following properties:

($F_I$) Every subset of X which contains a set of  $\mathfrak{F}$  belongs to  $\mathfrak{F}$.

($F_{II}$) Every finite intersection of sets of  $\mathfrak{F}$ belongs to  $\mathfrak{F}$.

Subsets in $\mathfrak{F}$ are called _neighbourhoods_ or _$\mathfrak{F}$-big_.

A mapping of underlying sets of filters is _continuous_ iff
the preimage of a neighbourhood is necessarily a neighbourhood.
Note that $\mathfrak{F}\cup \{\emptyset\}$ is a topology on $X$,
but this notion of continuity is stronger: it means continuous with dense image.

Let ዋ denote the category of filters and the continuous mappings of underlying sets.

## Idea

It is convenient to talk about _a filter on a set_
as _a filter structure on a set_
and think of it as _a space with a notion of smallness_,
 in analogy to a topological structure on a set
and thinking of it as
a (topological) space with a notion of nearness of points.

_A filter $\mathfrak{F}$ on a set $X$_ gives  a precise meaning
to phrases ``a property $P(x)$ holds for all points small enough'' and
``P holds for almost all elements of $X$'',
namely that
$ \{ x : P(x) \} \in \mathfrak{F} $,
in the same way that the topology (topological structure)
on a set gives
a precise meaning to the phrase
``a property $P(x)$ holds for all points sufficiently near a given point $x_0$'',
namely $\{ x : P(x) \} $ is a neighbourhood of $x_0$.

Thus, for example, in the neighbourhood filter of a point $x_0$
in a topological space,
a point $x$ is small iff $x$ is near $x_0$:
we think of $x$ near $x_0$ as an approximation to $x_0$
up to a small error.

A number of elementary notions can be formulated using the [[situs|category of simplicial objects of the category of filters]].

## Application to analysis and topology

In a [[topological space]] $X$, a filter $\mathfrak{F}$ on $X$ __[[convergence|converges]]__ to a point $x$ of $X$ if every [[neighbourhood]] of $x$ belongs to $\mathfrak{F}$, i.e. the map from $\mathfrak{F}$ to the neighbourhood filter of $x$ is continuous.  The filter on the empty set is the initial object; the proper filter on a single point is the terminal object.

The concepts of continuous function and uniformly continuous function may be defined quite nicely in terms of the continuity of maps of filters.
A map of topological spaces is continuous iff it induces continuous maps of neighbourhood filters. A map of uniform spaces is continuous iff it induces a continuous map of the uniform structures (filters on $M\times M$).

In a [[metric space]] $M$, a filter $\mathfrak{F}$ on $M$ is __Cauchy__ if it has elements of arbitrarily small diameter, i.e. the map $\mathfrak{F}\times \mathfrak{F}\to S\times S$ is continuous with respect to the uniformity (filter) on $S\times S$.  Then a [[sequence]] is a [[Cauchy sequence]] iff its eventuality filter is Cauchy, i.e. it induces a continuous map of certain filters.  This gives a diagram chasing meaning to the concept of completion of a metric space.

## See also

* [[filter]]


category of fractions

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea
 {#Idea}

In general, *[[localization]]* is a process of [[free construction|freely]] adjoining [[inverses]] to an algebraic structure. In [[category theory]], the _localization_ $\mathcal{C}[W^{-1}]$ of a [[category]] $\mathcal{C}$ at a [[class]] $W$ of its [[morphisms]] is -- if it exists -- the result of universally making all morphisms in $W$ into [[isomorphisms]].

Some authors call any localization of a category a _category of fractions_  ([Gabriel & Zisman, §1.1](#GabrielZisman67); [Borceux 1994 I, Def. 5.2.1](#Borceux94)), while other authors reserve this name for the case that $W$ admits a "[[calculus of fractions]]" (e.g. [Fritz 2011](#Fritz11)). Conversely, some authors use "localization" to refer exclusively to the latter case (e.g. [Verdier 1996, Def. 2.2.5](#Verdier96);  [Stacks Project](#StacksProject)).

Yet other authors take "localization" to mean specifically "[[reflective localizations]]", by default (e.g. [[Higher Topos Theory|Lurie 2009]]).

### Motivation

A classic example is the [[localization of a commutative ring]]: we can 'localize the ring $\mathbb{Z}$ away from the prime $2$' and obtain the ring $\mathbb{Z}[\frac{1}{2}]$, or localize it away from all primes and obtain its field of fractions: the field $\mathbb{Q}$ of [[rational number|rational numbers]].

The terminology is odd for historical and geometric reasons: localizing *at* a prime means inverting things *not* divisible by that prime, while inverting the prime itself is called localizing *away* from that prime.  The reason for this, as well as for the term 'localization', becomes more apparent when we consider examples of a more vividly geometric flavor.

For example, the ring $\mathbb{R}[x]$ consists of [[polynomial]] functions on the real line.  If we pick a point $a \in \mathbb{R}$ and localize $\mathbb{R}[x]$ by putting in an inverse to the element $(x-a)$, the resulting ring consists of [[rational function]]s defined _everywhere_ on the real line _except_ possibly at the point $a$.  This is called **localization _away from_ $a$**, or localization away from the [[ideal]] $I$ generated by $(x-a)$.

If on the other hand we put in an inverse to every element of $\mathbb{R}[x]$ that is _not_ in the ideal $I$, we obtain the ring of rational functions defined _somewhere_ on the real line _at least_ at the point $a$: namely, those without a factor of $(x-a)$ in the denominator.  This is called **localizing _at_ $a$**, or localizing at the ideal $I$.

Notice that what is literally 'localized' when localizing the ring is not the ring itself, but its [[spectrum of a commutative ring|spectrum]]: the spectrum becomes smaller.  The spectrum of $\mathbb{R}[x]$ is the whole real line.  When we localize away from $a$, the resulting ring has spectrum $\mathbb{R} - \{a\}$.  When we localize at $a$, the resulting ring has spectrum $\{a\}$.

The case of localizing $\mathbb{Z}$ can also be interpreted geometrically in a similar way, using [[scheme]] theory and [[arithmetic geometry]].

A ring is a very special case of a [[category]], namely a one-object [[Ab-enriched category]].  This article mainly treats the more general case of localizing an arbitrary category.  The localization of a category $C$ at a class of [[morphisms]] $W$ is the universal solution to making the morphisms in $W$ into [[isomorphisms]]; it is variously written $C[W^{-1}]$, $W^{-1}C$ or $L_W C$.  In some contexts, it also could be called the [[homotopy category]] of $C$ with respect to $W$.

## Definition

Let $C$ be a [[category]] and $W \subset Mor(C)$ a collection of [[morphisms]].

### General

A **localization** of $C$ by $W$ (or "at $W$") is

* a (generally [[large category|large]], see below) [[category]] $C[W^{-1}]$;

* and a [[functor]] $Q : C \to C[W^{-1}]$;

* such that

  * for all $w \in W$, $Q(w)$ is an [[isomorphism]];

  * for any category $A$ and any [[functor]] $F : C \to A$ such that $F(w)$ is an [[isomorphism]] for all $w \in W$, there exists a functor $F_W : C[W^{-1}] \to A$ and a [[natural transformation|natural isomorphism]] $F \simeq F_W \circ Q$;

  * the map between [[functor category|functor categories]]
  $$(-)\circ Q : Funct(C[W^{-1}], A) \to Funct(C,A)$$
  is [[fully faithful functor|full and faithful]] for every category $A$.

Note:

* if $C[W^{-1}]$ exists, it is unique up to [[equivalence]].

* In 2-categorical language, $C[W^{-1}]$ is the [[coinverter]] of the canonical natural transformation $s\to t$, where $s,t:W\to C$ are the "source" and "target" functors and $W$ is considered as a full subcategory of the [[arrow category]] $C ^{\mathbf{2}}$.

+-- {: .num_remark}
###### Remark
Contrary to what is sometimes asserted, the localization $C[W^{-1}]$ may generally be constructed when $C$ is large, even only in ZF where "large" means definable by class formulas. See Remark \ref{ZF} below. However, the localization might not be [[locally small category|locally small]], even if $C$ is. The tools of [[homotopy theory]], and in particular [[model category|model categories]], can be used to address this issue (see also at _[[homotopy category of a model category]]_).
=--

### Reflective localization

A special class of localizations are _[[reflective localizations]]_, those where the functor $C \to L_W C$ has a [[full and faithful functor|full and faithful]] [[right adjoint]] $L_W C \hookrightarrow C$.

In such a case

$$
  L_W C
    \stackrel{\overset{Q}{\leftarrow}}{\hookrightarrow}
  C
$$

this adjoint exhibits $L_W C$ as a [[reflective subcategory]] of $C$.

One shows that $L_W C$ is -- up to [[equivalence of categories]] -- the full subcategory on the $W$-[[local object]]s, and this property precisely characterizes such reflective localizations.

More on this is at _[[reflective localization]]_, _[[reflective subcategory]]_, _[[reflective sub-(∞,1)-category]]_, and [[reflective factorization system]].

### Localizations of enriched categories

Given a symmetric closed monoidal category $V$, a $V$-enriched category $A$ with underlying ordinary category $A_0$ and a subcategory $\Sigma$ of $A_0$
containing the identities of $A_0$, H. Wolff defines the corresponding theory of localizations. See [[localization of an enriched category]].

## Construction

There is a general construction of $C[W^{-1}]$, if it exists, which is however hard to use.  When the system $W$ has special properties, most notably when $W$ admits a [[calculus of fractions]] or a [[factorization system]], then there are more direct formulas for the [[hom-sets]] of $C[W^{-1}]$.

### General construction

Given $C$ a [[category]] and $W \,\subset\, Mor(C)$ is a set of [[morphisms]] (not necessarily forming a [[subcategory]]), we construct the localization of $C$. Let $W^{op}$ denote the corresponding set in the [[opposite category]] $C^{op}$.

Let $G$ be the following [[directed graph]]:

* the [[vertices]] of $G$ are the [[objects]] of $C$,

* the [[morphisms]] of $G$ between two vertices $x,y$ are given by the [[disjoint union]] $C(x,y)\coprod W^{op}(x,y)$.

The arrows in $W^{op}(x,y)$ are written as $\overline{f}$ for $f\in W(y,x)$.

Let $\mathcal{P}G$ be the [[free category]] on $G$. The [[identity morphisms]] are given by the empty path beginning and ending at a given object. We introduce a [[relation]] on the morphisms of $\mathcal{P}G$ and quotient by the [[equivalence relation]] $\sim$ generated by it to get $C[W^{-1}]$.

The equivalence relation $\sim$ is that generated by

* for all objects $x$ of $C$,

  $$(x;id_x;x) \sim (x;\emptyset;x)$$

* for all $f:x\to y$ and $g:y\to z$ in $C$,

  $$(x;f,g;z)\sim (x;g\circ f;z)$$

* for all $f:x\to y$ in $W$,

  $$(x;f,\overline{f};x)\sim (x;id_x;x)$$

  and

  $$(y;\overline{f},f;y)\sim (y;id_y;y)$$

+-- {: .num_remark #ZF}
###### Remark
With a little care, this general construction can be enacted even for [[large categories]] $C$, where "large" means given by a class formula in [[ZF]]. First, to define the free category $\mathcal{P} G$ on a large graph $G$, notice that an arrow of $\mathcal{P} G$ is a certain [[partial function]] or functional relation from $\mathbb{N}$ to $Arr(G)$ satisfying some conditions given by ZF formulas; each such relation is a "set" by the [[replacement axiom]], and the collection of such relations forms a definable class of sets. Second, in passing to the quotient, we remark that while the $\sim$-equivalence classes of morphisms in $\mathcal{P} G$ might be large (hence themselves could not be elements of a class), we can always use [[Scott's trick]] and consider instead those representatives in each equivalence class that have minimal ZF [[rank]] (using crucially the [[axiom of foundation]]); this collection of representatives is a *set*. Thus the collection of morphisms in the quotient may be identified with a definable class of sets.

The rub is that the localization might not be locally small, even if $C$ is.
=--

### Construction when there is a calculus of fractions

If the class $W$ admits a [[calculus of fractions]], then there is a simpler description of $C[W^{-1}]$ in terms of [[span|spans]] instead of zig-zags.  The idea is that any morphism $f: x \to z$ in $C[W^{-1}]$ is built from a morphism $f_2 : y \to z$ in $C$ and a morphism $f_1 : y \to x$ in $W$:

$$  x \stackrel{f_1}{\longleftarrow} y \stackrel{f_2}{\longrightarrow} z $$

For more on this, see the entry [[calculus of fractions]].

[[Dorette Pronk]] has extended this idea to construct a [[bicategory of fractions|bicategories of fractions]]  where a class of 1-arrows is sent to [[equivalence|equivalences]].

### In abelian categories

Localization is especially well developed in abelian setup where several competing formalisms and input data are used. See [[localization of an abelian category]].

## Properties
 {#Properties}

* The localization of the [[product category]] of two [[categories with weak equivalences]] is (if it exists) the product of their localizations

  $$
    (\mathcal{C} \times \mathcal{D})[ (W_{\mathcal{C}} \times W_{\mathcal{D}})^{-1} ]
    \;\;
      \simeq
    \;\;
    ( \mathcal{C}[W_{\mathcal{C}}^{-1}] )
     \times
    ( \mathcal{D}[W_{\mathcal{D}}^{-1}] )
    \,.
  $$

  This is because localization is a [[reflector]] into the [[exponential ideal]] of minimal categories with weak equivalences.

## Examples

* examples of localizations of [[locally small categories]] that are *not* themselves locally small:

  * [Krause 2008, Ex. 4.15](#Krause08)

  * [math.SE.a/703354](https://math.stackexchange.com/a/703354/58526)

## In higher category theory

The notion of localization of a category has analogs in [[higher category theory]].

For [[2-categories]]:

* [[2-localization]]

For [[(∞,1)-categories]] and the special case of reflective embeddings this is discussed in

* [[localization of an (∞,1)-category]].

Every [[locally presentable (∞,1)-category]] is presented by a [[combinatorial model category|combinatorial]] [[model category]]. Accordingly, there is a model for the localization of $(\infty,1)$-categories in terms of these models. This is called

* left [[Bousfield localization of model categories]]

See also [[localization of a simplicial model category]].

## Related concepts

* [[localizing subcategory]], [[localizer]]

* [[localization at an object]]

* [[reflective localization]]

  [[essential localization]]

  [[quintessential localization]]

  [[reflective bicategory]]

* [[monoidal localization]]

* [[simplicial localization]]

* [[localization of a simplicial model category]]

* [[localization of a ring]], [[localization of a commutative ring]]

  * [[localization of a module]]

* [[localization of model categories]]

  * [[localization of simplicial model categories]]

  * [[Bousfield localization of model categories]]

    * [[Bousfield localization of spectra]]

    * [[p-localization]]

* [[localization of a type at a family of functions]]

* [[colocalization]]

* [[ring of fractions]]

## References
 {#References}

Authors who say "category of fractions" for the result of inverting any class of morphisms:

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], §I.1 in: _[[Calculus of fractions and homotopy theory]]_, _Ergebnisse der Mathematik und ihrer Grenzgebiete_ **35**, Springer (1967)  [[doi:10.1007/978-3-642-85844-4](https://link.springer.com/book/10.1007/978-3-642-85844-4), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf)]

* {#Borceux94} [[Francis Borceux]], §5.2 of: _[[Handbook of Categorical Algebra]] vol. 1_, Cambridge UP (1994)

  > (See also vols. 2 and 3 for examples of the theory in action in abelian categories, sheaf theory etc.)

Authors who say "localization" for the result of inverting any class of morphisms:

* {#KashiwaraSchapira00} [[M. Kashiwara]], [[P. Schapira]], Section 7 of: _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften **332**, Springer (2006) [ [doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]

* Wikipedia, *[Localization of a category](https://en.wikipedia.org/wiki/Localization_of_a_category)*

and reserve "category of fractions" for the case that $W$ admits a [[calculus of fractions]]:

* {#Fritz11} [[Tobias Fritz]], *Categories of Fractions Revisited*, Morfismos **15**  2 (2011) 19-38 [[arXiv:0803.2587](https://arxiv.org/abs/0803.2587), [morfismos:vol15-n2-3](https://morfismos.math.cinvestav.mx/sites/default/files/Upload/vol15-n2-3.pdf)]

See also:

* {#Yekutieli19} [[Amnon Yekutieli]], Chapter 6 of: *Derived Categories*, Cambridge University Press (2019) [[doi:10.1017/9781108292825]( https://doi.org/10.1017/9781108292825), [arXiv:1610.09640](https://arxiv.org/abs/1610.09640)]

Authors who say "localization" for inverting with respect to a [[calculus of fractions]]:

* {#Verdier96} [[Jean-Louis Verdier]], Def. 2.2.5 in: _Des Catégories Dérivées des Catégories Abéliennes_, Astérisque **239** (1996) [[numdam:AST_1996__239__R1_0](http://www.numdam.org/issues/AST_1996__239__R1_0), [pdf](http://www.numdam.org/item/AST_1996__239__R1_0.pdf)]

* {#StacksProject} [[Stacks Project]], *4.27 Localization in categories* [[tag:04VB](https://stacks.math.columbia.edu/tag/04VB)]

With focus on localization of [[abelian categories]] and of [[triangulated categories]]

* [[Pierre Gabriel]], *[[Des catégories abéliennes]]*, Bulletin de la Société Mathématique de France, 90 (1962) 323-448 [[numdam:BSMF_1962__90__323_0](http://www.numdam.org/item?id=BSMF_1962__90__323_0)]

* [Verdier 1996](#Verdier96)

Review of localization in [[homotopy theory]]:

* [[William Dwyer]], _Localizations_, in: _Axiomatic, enriched and motivic homotopy theory_, NATO Sci. Ser. II **131**, Kluwer Acad. Publ. (2004) 3-28 [[doi:10.1007/978-94-007-0948-5](https://doi.org/10.1007/978-94-007-0948-5)]

For [[triangulated categories]]:

* {#Krause08} [[Henning Krause]], *Localization theory for triangulated categories*, in proceedings of *[Workshop on Triangulated Categories, Leeds 2006](https://www.commalg.org/2006/08/13/workshop-on-triangulated-categories-leeds/)* [[arXiv:0806.1324](https://arxiv.org/abs/0806.1324)]

A __terminological discussion__ prompted by question in which sense "localization" is a descriptive term or not is archived ion $n$Forum [here](https://nforum.ncatlab.org/discussion/481/localization/?Focus=23461#Comment_23461).

A formal implementation of Gabriel-Zisman localization in [[ZFC]], which in turn is implemented in the [[proof assistant]] [[Coq]] is in

* {#Simpson05} [[Carlos Simpson]], _Explaining Gabriel-Zisman localization to the computer_ ([arXiv:math/0506471](http://arxiv.org/abs/math/0506471), [web](http://math.unice.fr/~carlos/themes/verif.html))

A [[HoTT]]-[[Coq]]-formalization of left-exact [[reflective sub-(∞,1)-categories]] ([[localization of an (∞,1)-category]]) in [[homotopy type theory]] is in

* {#Shulman} [[Mike Shulman]], _[HoTT/Coq/Subcategories/LexReflectiveSubcategory.v](https://github.com/mikeshulman/HoTT/blob/master/Coq/Subcategories/LexReflectiveSubcategory.v)_

An original account of [[localization of commutative rings]] and of [[p-localization|p-local]] [[homotopy theory]] is

* {#Sullivan70} [[Dennis Sullivan]], _Localization, Periodicity and Galois Symmetry_ (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springer ([pdf](http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf))

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category of fractions > history

see at *[[category of fractions]]*