codomain fibration

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

For $C$ any [[category]], there is a [[functor]]

$$
  t \;\colon\; [I,C] \longrightarrow C
  \,,
$$

from the [[arrow category]] $[I,C] = Arr(C)$ that sends each [[morphism]] $(c_1 \stackrel{f}{\to} c_2) \in [I,C]$ to its [[codomain]] $c_2$.

This functor is always an [[Grothendieck opfibration]]. Under the [[Grothendieck construction]] it corresponds to the [[pseudofunctor]]

$$
  C_{(-)} \,\colon\, C \longrightarrow Cat
$$

that sends each object $c \in C$ to the [[overcategory]] $C_{/c}$.

If $C$ has all [[pullbacks]], then the functor is in addition a [[Grothendieck fibration]], hence a [[bifibration]].
Traditionally it is this fibered aspect is emphasised (and it even motivates the notion of *[[Cartesian fibrations]]*).  A [[right adjoint]] $u_*$ of $u^*$ exists for every morphism $u$ in $C$ iff C is a [[locally cartesian closed category]].

In any case, this functor $cod \colon [I,C] \to C$ is called the **codomain fibration** of $C$.

Some say **basic fibration** or **[[indexed category|self-indexing]]** or the **fundamental fibration** --- anything with so many names must be important!

If instead of the codomain the domain is used, one obtains the dual notion: [[domain opfibration]].

## Details

We spell out the details of the functor, of its cartesian and opcartesian morphisms and their properties.

### The arrow category

Recall from the discussion at [[arrow category]] that the objects in $Arr(C)$ are morphisms in $C$ and the morphisms $(f:x_1\to x_2)\to (g: y_1\to y_2)$ in $Arr(C)$ are the commutative squares in $C$ of the form

$$\array{
    x_1 &\stackrel{v}\longrightarrow& y_1
    \\
    \big\downarrow\mathrlap{f}
    &&
    \big\downarrow\mathrlap{g}
    \\
    x_2 &\stackrel{u}\longrightarrow& y_2
}$$

with the obvious composition.

### The functor

The [[functor]] $ cod : Arr(C)\to C$ is given on objects by the codomain (= target) map, and on morphisms it gives the lower arrow of the commutative square.

$$
  cod :
  \;\;
  \left(
  \array{
    x_1 &\stackrel{v}\to& y_1
    \\
    \downarrow\mathrlap{f}
    &&
   \downarrow\mathrlap{g}
   \\
   x_2 &\stackrel{u}\to& y_2
  }
  \right)
  \;\;
  \mapsto
  \;\;
  (x_2 \stackrel{u}\to y_2)
  \,.
$$

If we write $[I,C]$ for the [[arrow category]], where $I$ is the [[interval category]] $I = \{a \to b\}$, then this functor is the [[hom-functor]] applied to the inclusion $\iota_1 : {b} \to \{a \to b\}$

$$
  cod = Hom_\text{Cat}(\iota_1, 1_C) : [I,C] \to [{*}, C] = C
  \,.
$$

### The op-cartesian lifts

That the functor $cod : [I,C] \to C$ is an [[Grothendieck fibration|opfibration]] means that for each object $\hat c_1 \to c_1$ of $[I,C]$, each morphism $c_1 \stackrel{f}{\to} c_2$ in $C$ has a lift to a morphism

$$
  \array{
     \hat c_1 &\to& ??
     \\
     \downarrow && \downarrow
     \\
     c_1 &\to& c_2
  }
$$

in $[I,C]$ that is a [[cartesian morphism|opcartesian morphism]].

Such a lift is given by

$$
  \array{
     \hat c_1 &\stackrel{Id}{\to}& \hat c_1
     \\
     \downarrow && \downarrow
     \\
     c_1 &\to& c_2
  }
  \,.
$$

For given any commuting triangle

$$
  \array{
     && c_2
     \\
     & \nearrow && \searrow
     \\
     c_1 &&\to&& c_3
  }
$$

in $C$, and any lift

$$
  \array{
    \hat c_1 &\to& d
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_3
  }
$$

of $c_1 \to c_3$, there is the unique lift

$$
  \array{
    \hat c_1 &\to& d
    \\
    \downarrow && \downarrow
    \\
    c_2 &\to& c_3
  }
$$

such that

$$
  \left(
  \array{
    \hat c_1 &\stackrel{Id}{\to}& \hat c_1
    &\to& d
    \\
    \downarrow && \downarrow && \downarrow
    \\
    c_1 &\to& c_2 &\to& c_3
  }
  \right)
  \;\;\;
  =
  \;\;\;
  \array{
    \hat c_1
    &\to& d
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_3
  }
  \,.
$$

### The cartesian lifts

If $C$ has [[pullback]]s, then $cod : [I,C] \to C$ is in addition a [[Grothendieck fibration|fibered category]] in the sense of Grothendieck:

for every object $\hat c_2 \to c_2$ in $[I,C]$, the cartesian lift of a morphism $c_1 \to c_2$ in $C$ is given by the morphism

$$
  \array{
    c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_2
  }
  \,.
$$

Because for

$$
  \array{
     && c_3
     \\
     & \swarrow && \searrow
     \\
     c_1 &&\to&& c_2
  }
$$

any commuting triangle in $C$, and for

$$
  \array{
    d &\to& \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_3 &\to& c_2
  }
$$

any lift of $c_3 \to c_2$ in $[I,C]$, which by the commutativity of the triangle we may write as

$$
  \array{
    d &\to& &\to& \hat c_2
    \\
    \downarrow && && \downarrow
    \\
    c_3 &\to& c_1 &\to& c_2
  }
$$

there is, precisely by the universal property of the [[pullback]], a unique morphism, $d\to c_1 \times_{c_2} \hat c_2$ in $C$ such that this factors as

$$
  \array{
    d &\to& c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
    \\
    \downarrow && \downarrow && \downarrow
    \\
    c_3 &\to& c_1 &\to& c_2
  }
  \,.
$$

### Direct image operation

Recall that in an  [[Grothendieck fibration|opfibration]] $p : E\to B$ , the _direct image_ $f_!$ of an object $e \in E$ along a morphism $p(e) \to d$ is the codomain $f_!(e)$ of [[generalized the|the]] opcartesian lift $\hat f : e \to f_! e$ of $f$.

By the above discussion this means that in the codomain opfibration $cod : [I,C] \to C$ the direct image of an object $\hat c_1 \to c_1$ in $[I,C]$ along some morphism $f : c_1 \to c_2$ is the composite morphism $\hat c_1 \to c_1 \to c_2$ in $C$, regarded as an object in $[I,C]$: this yields the functor

$$
  f_! : C/{c_1} \to C/{c_2}
$$

of [[overcategory|overcategories]] obained by postcomposition with $f$.

### Inverse image operation

Recall that in an  [[Grothendieck fibration|fibration]] $p : E\to B$ , the _inverse image_ $f^*$ of an object $e \in E$ along a morphism $d \to p(e) $ is the domain $f^*(e)$ of [[generalized the|the]] cartesian lift $\hat f : f^* e \to e$ of $f$.

By the above discussion this means that in the codomain fibration $cod : [I,C] \to C$ the inverse image of an object $\hat c_2 \to c_2$ in $[I,C]$ along some morphism $f : c_1 \to c_2$ is the morphism out of the [[pullback]] $f^* c_2 = c_1 \times_{c_2} \hat c_2 \to c_1$ in $C$, regarded as an object in $[I,C]$: this yields the functor

$$
  C/{c_1} \leftarrow C/{c_2} : f^*
$$

of [[overcategory|overcategories]] obained by pullback.

### Adjointness of direct and inverse image

For every morphism $f : c_1 \to c_2$ in $C$, the direct and inverse image functors are a pair of [[adjoint functor]]s

$$
  f_! : C/{c_1} \to C/{c_2} : f^*
$$

with $f_!$ [[left adjoint]] and $f^*$ [[right adjoint]], $f_! \dashv f^*$.

By the above discussion, the adjunction isomorphism

$$
  Hom_{C_2}(f_! \hat c_1, \hat c_2)
  \simeq
  Hom_{C_1}(\hat c_1, f^*\hat c_2)
$$

is given by the universal property of the [[pullback]] operation, which says that morphisms

$$
  (f_! \hat c_1 \to \hat c_2)
  =
  \left(
  \array{
    \hat c_1 &\to& \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_2
  }
  \right)
$$

factor uniquely through the [[pullback]]

$$
  \left(
  \array{
    \hat c_1 &\to& c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
    \\
    &\searrow & \downarrow && \downarrow
    \\
    && c_1 &\to& c_2
  }
  \right)
$$

and hence uniquely correspond to morphisms

$$
  (\hat c_1 \to f^* \hat c_2)
  =
  \left(
  \array{
    \hat c_1 &\to& c_1 \times_{c_2} \hat c_2
    \\
    \downarrow && \downarrow
    \\
    c_1 &\to& c_2
  }
  \right)
  \,.
$$

If $C$ is a [[model category]], and $u:c\to d$ a morphism in $C$, we can consider the induced model structure on the overcategories $C/c$, and $C/d$. The adjoint pair

$$
   u_! : C/c \leftrightarrows  C/d : u^*
$$

is then a Quillen pair.

## Monadic descent

Since the codomain fibration $cod : [I,C] \to C$ is a [[bifibration]] when $C$ has all [[pullback]]s, there is a notion of [[monadic descent]] in this case. Details on this are at [monadic descent for codomain fibrations](http://ncatlab.org/nlab/show/monadic+descent#ForCodomainFibs).

## The Subobject Fibration
 {#TheSubobjectFibration}

By restricting our attention to a [[subset]] of [[morphisms]] in the codomain fibration and using the notion of the [[skeleton of a fibration]], we may define a fibration on a category $\mathcal{C}$ with pullbacks called the *subobject fibration* whose fibers are categories of [[subobjects]] for objects of $\mathcal{C}$.

Beginning with the codomain fibration $cod \colon \mathcal{C}^\to \longrightarrow \mathcal{C}$ on a category $\mathcal{C}$ with [[pullbacks]] (now writing $C^\to$ for the [[arrow category]]), we restrict our attention to the subcategory
$$
Mono(\mathcal{C})\subseteq\mathcal{C}^\to,
$$
the full subcategory of $\mathcal{C}^\to$ whose objects are monomorphisms in $\mathcal{C}$, called the **monomorphism category of $\mathcal{C}$**. The resulting functor
$$
cod:Mono(\mathcal{C})\to\mathcal{C}
$$
is again a fibration since monomorphisms are stable under pullback; we will call this the **monomorphism fibration of $\mathcal{C}$**. The fibers $Mono(\mathcal{C})_X$ for $X\in\mathcal{C}$ are [[thin categories]] since parallel monos in a slice category are equal, but they aren't subobject categories since antisymmetry is only weakly satisfied -- objects with antiparallel arrows between them are necessarily isomorphic, but not necessarily equal. To remedy this, we take the [[fibered skeleton]] of the monomorphism fibration; briefly, we convert it into an [[indexed category]] using the [[Grothendieck construction]], take the [[skeleton]] of each index category, then turn it back into a fibration using the Grothendieck construction in the other direction. The resulting fibration is denoted
$$
cod:Sub(\mathcal{C})\to\mathcal{C}
$$
and called the **subobject fibration of $\mathcal{C}$**, and the fibers $Sub(\mathcal{C})_X$ are skeletal thin categories, also known as [[poset categories]].

If we take $\mathcal{C}=Set$ then the fibers $Mono(Set)_X$ of the monomorphism fibration are proper classes consisting of all sets isomorphic to subsets of $X$, which isn't what we want. The fibers $Sub(Set)_X$ consist of one representative from each isomorphism class of sets isomorphic to subsets of $X$, and is thusly isomorphic to the powerset of $X$ viewed as a poset. That is, $Sub(Set)_X\cong\mathcal{P}(X)$ as posets, with equality holding if we choose the right representatives.

## In higher category theory

We discuss the codomain fibration in [[higher category theory]].

### In 2-category theory

A categorification in dimension 2  (see [[2-category theory]]) is a codomain 2-fibration, whose main example is $Cat^2$ over $Cat$.

+--{: .query}
[[Mike Shulman]]: I still don't believe that that is a 2-fibration.  How do you lift the 2-cells?

[[David Roberts]]: How does one lift the 2-cells in a 2-fibration anyway? The case of $Cat^\mathbf{2}\to Cat$ (using weak 2-functors in $Cat^\mathbf{2}$) should in my opinion be an guiding example for this. Although, perhaps it would be better to consider (at least at first) the underlying (2,1)-category or even the (2,1)-category $Gpd$.

[[Mike Shulman]]: I think the guiding example of a 2-fibration should actually be $Fib \to Cat$, as in [Hermida's paper](http://www.cs.math.ist.utl.pt/s84.www/cs/claudio/articles/2-fib.ps).  There, you can lift the 2-cells, because in each fibration you can lift the 1-cells.
=--

### In $(\infty,1)$-category theory
 {#InInfinityCategoryTheory}

Let $\mathcal{X}$ be an [[(∞,1)-category]].

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

The codomain fibration

$$
  Cod : \mathcal{X}^I \to \mathcal{X}
$$

is an [[coCartesian fibration]].

It is classified under the [[(∞,1)-Grothendieck construction]] by

$$
  A \mapsto \mathcal{X}_{/A}
  \,,
$$

where on the right we have the [[over-(∞,1)-category]].

=--

This is a special case of ([Lurie, corollary 2.4.7.12](#Lurie)).

For $\mathcal{X}$ an [[(∞,1)-topos]], this is the canonical [[(infinity,2)-sheaf]].

## As a universe

(...)

$\mathbf{H}$ an [[(∞,1)-topos]] the codomain fibration is the [[dependent sum]]

$$
  \sum_{Type} : \mathbf{H}_{/Type} \to \mathbf{H}_{/*} \simeq \mathbf{H}
$$

where $Type \in \mathbf{H}$ is the [[object classifier]], of some size. This is the internal [[universe]]. Since the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ is the [[context]] given by $X$, in a precise sense
$\mathbf{H}_{/Type}$ is the "context of the universe". And so this says that the codomain fibration is the "context of the universe" regarded over the base $\infty$-topos which is the "outermost universe".

(...)

## Related concepts

* [[fibration of points]]

* [[tangent category]]

* [[tangent (infinity,1)-category]]

## References

* {#Lurie} [[Jacob Lurie]], Section 2.4.7 of: _[[Higher Topos Theory]]_

[[!redirects codomain fibrations]]

[[!redirects codomain opfibration]]
[[!redirects codomain opfibrations]]

[[!redirects self-indexing]]
[[!redirects self-indexing of a category]]

[[!redirects family fibration]]
[[!redirects family fibrations]]


Cody Roux

[[!redirects Roux Cody]]

* [webpage](https://www.andrew.cmu.edu/user/croux/)

## related $n$Lab entries

* [[pure type system]]

category: people


coefficient

# Contents
* table of contents
{: toc}

## Definition

There are two different meanings of the term _coefficient_ in mathematics, one

* [In algebra and analysis](#InAlgebraAndAnalysis)

and one

* [In cohomology and homology](#InCohomologyAndHomology).

The two notion do however coincide for [[ordinary homology]]/[[ordinary cohomology]] expressed as [[singular homology]]/[[singular cohomology]]. See remark \ref{RelationInSingularCoHomology} below.

### In algebra and analysis
 {#InAlgebraAndAnalysis}

In [[algebra]] and [[analysis]], a _coefficient_ is an element of a [[ring]] (or [[rig]]) $R$ that appears in scalar multiplication; more generally, _coefficients_ are elements of $R$ that appear in a [[linear combinations]].  Thus, we multiply a coefficient by an element of an $R$-[[module]] $M$ (which may even be an algebra, [[associative algebra|associative]] or [[nonassociative algebra|not]]) to get another element of $M$.

For example, given a [[polynomial]] $P = \sum_n a_n x^n$ over $R$ in the [[variable]] $x$, each $a_n \in R$ is the coefficient on $x^n$ in $P$.  The module $M$ here is the [[symmetric algebra]] over $R$.

### In cohomology and homology
 {#InCohomologyAndHomology}

In [[cohomology]] _coefficients_ are what the cohomology takes values in. For [[ordinary cohomology]] $H^\bullet(-,A)$ the [[abelian group]] $A$ is the _coefficient group_. For [[generalized (Eilenberg-Steenrod) cohomology]] $H^\bullet(-,E)$ the given [[spectrum]] $E$  that [[Brown representability|represents]] it is the _coefficient spectrum_. Dually for [[homology]].

It is this notion of "coefficient" that appears in terms like

* _[[universal coefficient theorem]]_

* _[[local system of coefficients]]_

* _[[local coefficient bundle]]_.

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

This cohomology-theoretic usage of "coefficient" is almost certainly originally derived from the [former](#InAlgebraAndAnalysis) one.  When [[ordinary homology]] with coefficients in $A$ (in the homological sense) is defined using [[singular homology|singular]] or [[cellular homology|cellular]] [[chains]], the chains themselves are [[formal linear combinations]] of [[singular simplices]] or cells whose coefficients (in the algebraic sense) are elements of $A$.  For more general homology and cohomology theories, however, the "coefficient object" is no longer directly interpretable using coefficients in the algebraic sense.

=--

category: disambiguation

[[!redirects coefficient]]
[[!redirects coefficients]]


coend > history

< [[coend]]
[[!redirects coend -- history]]


coend in a derivator

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

# Ends and coends in a derivator
* table of contents
{: toc}

## Idea

[[end|Ends]] and [[coends]] are special sorts of [[limit]] and [[colimit]], respectively, and have corresponding sorts of [[homotopy limits]] and colimits -- [[homotopy ends]] and coends.  Since a [[derivator]] is a formal structure for computing homotopy limits and colimits, there are corresponding notions of ends and coends in a derivator.

## Definitions

Let $D$ be a [[derivator]] indexed on a 2-category $Dia$ of diagram shapes, let $A\in Dia$, and let $H\in D(A^{op}\times A)$; we wish to define the *coend* of $H$ (obvious dualizations will yield its *end*).  We will give four equivalent definitions, each of which generalizes a classical construction of ordinary coends in terms of colimits.

### As a colimit over simplices

One classical construction of a coend as a colimit proceeds by constructing an auxiliary category $A^{tw}$, whose objects are the objects and arrows of $A$, with morphisms from each arrow of $A$ (regarded as an object of $A^{tw}$) to its domain and codomain.  There is a functor $p\colon A^{tw}\to A^{op}\times A$ which sends each object $x$ to $(x,x)$ and each arrow $f\colon x\to y$ to $(y,x)$, and the coend of $H\colon A^{op}\times A\to C$ can be constructed as the colimit of $p^* H\colon A^{tw}\to C$.

We can mimic this in a derivator, except that we need to "homotopify" $A^{tw}$ by including higher information as well.  Thus, let $(\Delta / A)^{op}$ denote the opposite of the category of simplices of $A$.  Thus its objects are functors $x\colon [n]\to A$, and its morphisms from $x\colon [n]\to A$ to $y\colon [m]\to A$ are functors $[m]\to [n]$ making the evident triangle commute.  There is a functor $p\colon (\Delta / A)^{op} \to A^{op}\times A$ which sends $x\colon [n]\to A$ to $(x(n), x(0))$.

Note that $A^{tw}$, as constructed above, is the full subcategory of $(\Delta / A)^{op}$ containing only the 0-simplices and 1-simplicies.  The inclusion of this subcategory is [[final functor|final]], but not [[homotopy final functor|homotopy final]].  Thus, for ordinary colimits it suffices to consider $A^{tw}$, but for homotopy colimits we need all of $(\Delta / A)^{op}$.

Hence, we define the **(homotopy) coend** of $H\in D(A^{op}\times A)$ to be the (homotopy) colimit of $p^* H \in D((\Delta / A)^{op})$.

### As a colimit over arrows

Another classical construction of a coend as a colimit involves a different auxiliary category, the opposite [[twisted arrow category]] $Tw(A)^{op}$.  The objects of $Tw(A)^{op}$ are arrows $f\colon x\to y$ in $A$, and its morphisms from $f\colon x\to y$ to $g\colon z\to w$ are commutative squares
$$ \array{
  x & \to & z\\
  ^f \downarrow & & \downarrow^g\\
  y & \leftarrow & w }
$$
in $A$.  There is a projection
$$ r \colon Tw(A)^{op} \to A^{op}\times A $$
sending $f\colon x\to y$ to $(y,x)$, and the coend of $H\colon A^{op}\times A\to C$ can be constructed as the colimit of $r^* H\colon Tw(A)^{op} \to C$.

Amazingly, this version needs no modification to become homotopical.  Given $H\in D(A^{op}\times A)$ in a derivator, we can simply restrict along $r$ to $Tw(A)^{op}$, then take the (homotopy) colimit.  To see that this agrees with the previous definition, it suffices to factor $q$ through $r$ via a [[homotopy final functor]] $s$:
$$ \array{ & & Tw(A)^{op} \\
  & ^{s}\nearrow & & \searrow^{r}\\
  (\Delta / A)^{op} & & \underset{q}{\to} & & A^{op}\times A } $$
The definition of $s$ is simple: we regard an $n$-simplex as a string of $n$ composable arrows in $A$ and take its composite.  The morphisms in the two categories match nicely.  To show that $s$ is homotopy final, we must show that
$$\array{(\Delta / A)^{op} & \xrightarrow{s} & Tw(A)^{op} \\
  \downarrow & & \downarrow \\
  * & \to & * }$$
is a [[homotopy exact square]].  For this, it suffices to show that it becomes homotopy exact when pasted with any [[comma square]]
$$\array{Q_{f} & \overset{}{\to} & *\\
  \downarrow & \swArrow & \downarrow^{f}\\
  (\Delta / A)^{op}& \underset{}{\to} & Tw(A)^{op}.}$$
The objects of the category $Q_{f}$ are strings of composable $n\ge 2$ arrows whose composite is $f$.  Its morphisms are like those of $(\Delta / A)^{op}$, but the first and last face maps are also given by composition instead of forgetting.  In fact, it is precisely the category of simplices of the fiber of the [[two-sided bar construction]] $B(A(-,b),A,A(a,-))$ over $f$.

However, the simplicial map $B(A(-,b),A,A(a,-)) \to A(a,b)$, with $A(a,b)$ a discrete simplicial set, is well-known to be a [[simplicial homotopy equivalence]] and thus a weak equivalence of simplicial sets.  Thus, each of its fibers is simplicially contractible, and hence each $Q_f$ has contractible nerve.  This implies that
$$ \array{ Q_f & \to & * \\ \downarrow & & \downarrow \\ * & \to & * } $$
is homotopy exact, and thus $s$ is homotopy final.

### As a bar construction

Another classical construction of a coend as a colimit is as the [[coequalizer]] of a [[parallel pair]]
$$ \coprod_{f\colon x\to y} H(y,x) \;\rightrightarrows\; \coprod_{x} H(x,x). $$
This can be obtained in a straightforward way from the previous construction.  If $Ppr$ denotes the [[walking structure|walking]] parallel pair $(1 \rightrightarrows 0)$, then there is a functor $q\colon A^{tw}\to Ppr$ sending each object of $A$ to 0, each arrow of $A$ to 1, and sorting the morphisms by whether they map an arrow to its domain or to its codomain.

Then the above parallel pair is the (pointwise) [[left Kan extension]] of $p^* H \colon A^{tw} \to C$ along $q$.  Because $q$ is a [[discrete opfibration]], left Kan extension along it can be computed with colimits over its fibers -- since each fiber is discrete, we obtain the coproducts above.  And since the colimit of $p^*H$ is equivalently its left Kan extension to the [[point]], the functoriality of Kan extensions means that $\colim^{A^{tw}} p^* H$ is isomorphic to $\colim^{Ppr} Lan_q p^* H$, the latter being precisely the above coequalizer.

We can homotopify this in a straightforward way as well.  Let $p\colon (\Delta / A)^{op} \to A^{op}\times A$ be as above, and let $q\colon (\Delta / A)^{op} \to \Delta^{op}$ be the obvious forgetful functor (whose target is the opposite of the [[simplex category]]).  Note that as before, $Ppr$ is a final, but not homotopy-final, subcategory of $\Delta^{op}$.  The functoriality of homotopy Kan extensions in a derivator means that the homotopy colimit of $p^* H\in D((\Delta / A)^{op})$ can equivalently be calculated as the homotopy colimit of $q_! p^* H \in D(\Delta^{op})$.

Note that since an object $D(\Delta^{op})$ is a [[simplicial object]] of $D$, it makes sense to call its colimit [[geometric realization]].  Moreover, the homotopy version of $q$ is also a discrete opfibration, and since pullbacks of fibrations are [[homotopy exact square|homotopy exact]], homotopy Kan extensions along $q$ are also computed as colimits over its fibers.  These fibers are also discrete, so we obtain a simplicial diagram of the following sort:

$$ \cdots \coprod_{x_0\xrightarrow{f_1} x_1 \xrightarrow{f_2} x_2} H(x_2,x_0)
\underoverset{\to}{\to}{\underoverset{\leftarrow}{\leftarrow}{\to}}
\coprod_{x_0\xrightarrow{f_1} x_1} H(x_1,x_0)
\underoverset{\to}{\to}{\leftarrow}
\coprod_{x} H(x,x)
$$

This is a derivator version of the [[bar construction]] of $H$.  (A bar construction is perhaps the most classical construction of homotopy coends.)

### As a weighted colimit

A last classical construction of a coend is as a [[weighted colimit]] of $H\colon A^{op}\times A\to C$, weighted by the [[hom-functor]] $hom\colon A^{op}\times A \to Set$.

In order to homotopify this, recall that weighted colimits can be constructed in terms of Kan extensions by first Kan extending to the [[collage]] of the weighting (pro)functor, then restricting to the target object.  Thus, let $B$ denote the collage of $hom\colon A^{op}\times A \to Set$ regarded as a profunctor $1 ⇸ A^{op}\times A$, with inclusions $u\colon 1\to B$ and $v\colon A^{op}\times A \to B$.  We can therefore define the **homotopy coend** of $H\in D(A^{op}\times A)$ to be $u^* v_! H$.

To show that this is the same as the previous definitions, we simply observe that there is a comma square
$$\array{Tw(A)^{op}& \overset{r}{\to} & A^{op}\times A\\
  ^!\downarrow & \swArrow_\alpha & \downarrow^v\\
  1& \underset{u}{\to} & B}
$$
Thus, by one of the axioms of a derivator, $u^* v_! H \cong \colim (r^* H)$.

[[!redirects end in a derivator]]
[[!redirects ends in a derivator]]
[[!redirects ends in derivators]]
[[!redirects coend in a derivator]]
[[!redirects coends in a derivator]]
[[!redirects coends in derivators]]


coequaliser > history

< [[coequaliser]]
[[!redirects coequaliser -- history]]


coequalizer

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###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _coequalizer_ in a general [[category]] is the generalization of the construction where for two [[functions]] $f,g$ between [[sets]] $X$ and $Y$
$$X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} Y$$
one forms the set $Y/_\sim$ of [[equivalence classes]] induced by the [[equivalence relation]] generated by the relation
$$f(x)\sim g(x)$$
for all $x \in X$. This means that the [[quotient]] function $p \colon Y \longrightarrow Y/_\sim$ satisfies
$$p \circ f = p \circ g$$
(a map $p$ satisfying this equation is said to "co-equalize" $f$ and $g$) and moreover $p$ is [[universal property|universal]] with this property.

In this form this may be phrased generally in any [[category]].

## Definition

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

In some [[category]] $\mathcal{C}$, the **coequalizer** $coeq(f,g)$ of two [[parallel morphisms]] $f$ and $g$ between two [[objects]] $X$ and $Y$ is (if it exists), the _[[colimit]]_ under the [[diagram]] formed by these two morphisms
$$\array{
    X
    &&
      \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}
    &&
    Y
    \\
    & \searrow && \swarrow_{\mathrlap{p}}
    \\
    && coeq(f,g)
  }.$$

=--

Equivalently:

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

In a [[category]] $\mathcal{C}$ a [[diagram]]
$$X \underoverset{g}{f}{\rightrightarrows} Y \overset{p}{\rightarrow} Z$$
is called a **coequalizer** diagram if

1. $p \circ f = p \circ g$;

1. $p$ is [[universal property|universal]] for this property: i.e. if $q \colon  Y \to W$ is a morphism of $\mathcal{C}$ such that $q \circ f = q \circ g$, then there is a unique morphism $q' \colon Z \to W$ such that $q' \circ p = q$
$$
  \array{
     X
       &\stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}&
     Y
      &\overset{p}{\longrightarrow}&
     Z
     \\
     && {}^{\mathllap{q}}\downarrow & \swarrow_{\mathrlap{\exists ! \, q'}}
     \\
     && W
  }
$$

=--

\begin{remark}
By [[formal duality]], a coequalizer in $\mathcal{C}$ is equivalently an [[equalizer]] in the [[opposite category]] $\mathcal{C}^{op}$.
\end{remark}

\begin{remark}
  Morphisms that coequalize some pair of [[parallel morphisms]] are called *[[regular epimorphisms]]*.
\end{remark}

## Properties

### Relation to kernel pairs
 {#RelationToKernelPairs}

\begin{proposition}
In any category:

* If a [[morphism]]

  1. is the [[coequalizer]] of some pair of [[parallel morphisms]] (hence: is a [[regular epimorphism]])

  1. has a [[kernel pair]],

  then it is also the coequalizer of its kernel pair.

* If a [[kernel pair]] has a coequalizer, then it is the kernel pair of its coequalizer.

\end{proposition}
(e.g. [Borceux 1994, Prop. 2.5.7, 2.5.8](#Borceux94), [Taylor 1999, Lemma 5.6.6](#Taylor99)

### Relation to pushouts
 {#RelationToPushout}

Coequalizers are closely related to [[pushouts]]:

\begin{proposition}
\label{CoequalizedAsAPushout}
A diagram
$$
  X
  \underoverset
    {f}
    {g}
    {\rightrightarrows}
  Y
   \overset{p}{\longrightarrow}
  Z
$$
is a coequalizer diagram, def. \ref{CoequalizerDiagram}, precisely if
$$\array{
     X \sqcup X
     &\overset{(f,g)}{\longrightarrow}& Y
     \\
     \big\downarrow && \big\downarrow{^\mathrlap{p}}
     \\
     X &\underset{}{\longrightarrow}& Z
  }
$$
is a [[pushout]] diagram.
\end{proposition}

Conversely:

\begin{proposition}
\label{PushoutAsACoequalizer}
$$
\array{
    A &\overset{f_1}{\longrightarrow}& B
    \\
    {\mathllap{{}^{f_2}}}\Big\downarrow
    &&
    \Big\downarrow{{}^\mathrlap{p_1}}
    \\
    C &\underset{p_2}{\longrightarrow}& D
  }$$
is a [[pushout]] square, precisely if
$$
  A
   \underoverset
     {
       q_2
         \circ
       f_2
     }
     {
       q_1
         \circ
       f_1
    }
     {\rightrightarrows}
  B \sqcup C
  \overset{(p_1,p_2)}{\longrightarrow}
  D
$$
is a coequalizer diagram.
\end{proposition}
(Here $B \overset{q_1}{\to} B \sqcup C \overset{q_2}{\leftarrow} C$ denotes the two [[coprojections]] into the [[coproduct]].)

## Examples

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

For $\mathcal{C} =$ [[Set]], the coequalizer of two [[functions]] $f$, $g$ is the [[quotient set]] by the [[equivalence relation]] generated by the [[relation]] $f(x) \sim g(x)$ for all $x \in X$.

=--

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

For $\mathcal{C} =$ [[Top]], the coequalizer of two [[continuous functions]] $f$, $g$ is the [[topological space]] whose underlying set is the [[quotient set]] from example \ref{QuotientSet}, and whose topology is the corresponding [[quotient topology]].

=--

## Related concepts

* [[equalizer]]

* [[regular epimorphism]]

* [[propositional truncation object]]

## References

Coequalizers were defined in the paper

* {#EH} [[Beno Eckmann]], [[Peter J. Hilton]], _Group-like structures in general categories II.  Equalizers, limits, lengths_.  Mathematische Annalen 151:2 (1963), 150–186.  [doi:10.1007/bf01344176](https://doi.org/10.1007/bf01344176).

for any finite collection of parallel morphisms.
The paper refers to them as _right equalizers_, whereas [[equalizers]] are referred to as _left equalizers_.

Textbook accounts:

* {#Borceux94} [[Francis Borceux]], Section 2.4 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

* {#Taylor99} [[Paul Taylor]], *[[Practical Foundations of Mathematics]]*, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999 ([webpage](http://www.paultaylor.eu/~pt/prafm/index.html))

[[!redirects coequalizers]]
[[!redirects coequaliser]]
[[!redirects coequalize]]
[[!redirects coequalise]]
[[!redirects coequalisers]]
[[!redirects coequalizes]]
[[!redirects coequalises]]


coequalizer type

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### Context
#### Type theory
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[[!include type theory - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The instantiation of a [[coequalizer]] as a type in [[dependent type theory]].

## Definition

In [[dependent type theory]], given types $A$ and $B$ and functions $f:A \to B$ and $g:A \to B$, the **coequalizer type** or **coequaliser type** of $f$ and $g$ is a [[higher inductive type]] $\mathrm{coeq}_{A,B}(f, g)$. There are two different way to define the coequalizer type. The first way states that $\mathrm{coeq}_{A,B}(f, g)$ is generated by the constructors

$$\mathrm{in} \colon B \to \mathrm{coeq}_{A,B}(f, g)$$

$$\mathrm{glue} \colon \prod_{x:A}in(f(x)) =_{\mathrm{coeq}_{A,B}(f, g)} in(g(x))$$

The second way states that $\mathrm{coeq}_{A,B}(f, g)$ is generated by

$$\mathrm{in} \colon B \to \mathrm{coeq}_{A,B}(f, g)$$

$$\mathrm{glue} \colon \prod_{x:B} \prod_{y:B} \prod_{z:A} (f(z) =_B x) \to (g(z) =_B  y) \to (\mathrm{in}(x) =_{\mathrm{coeq}_{A,B}(f, g)} \mathrm{in}(y))$$

### Induction and recursion principles

The recursion principle for coequalizer types says that given a type $C$ together with

* a function $c_\mathrm{in}:B \to C$
* a dependent function $c_\mathrm{glue}:\prod_{x:B} \prod_{y:B} \prod_{z:A} (f(z) =_B x) \to (g(z) =_B y) \to (c_\mathrm{in}(x) =_C c_\mathrm{in}(y))$

there exists a function $c:\mathrm{coeq}_{A, B}(f, g) \to C$ such that

* for all $x:B$, $c(\mathrm{in}(x)) \equiv c_\mathrm{in}(x)$, and
* for all $x:B$, $y:B$, $z:A$, $p:f(z) =_B x$, and $q:g(z) =_B y$,
$$\mathrm{ap}_c(\mathrm{glue}(x,y,z,p,q)) \equiv c_\mathrm{glue}(x, y, z, p, q)$$

Similarly, the induction principle says that given a type family $C(w)$ indexed by $w:\mathrm{coeq}_{A, B}(f, g)$ together with

* a dependent function $c_\mathrm{in}:\prod_{x:B} C(\mathrm{in}(x))$
* a dependent function $c_\mathrm{glue}:\prod_{x:B} \prod_{y:B} \prod_{z:A} \prod_{p:f(z) =_B x} \prod_{q:g(z) =_B y} c_\mathrm{in}(x) =_C^{\mathrm{glue}(x,y,z,p,q)} c_\mathrm{in}(y)$

there exists a dependent function $c:\prod_{z:\mathrm{coeq}_{A, B}(f, g)} C(z)$ such that

* for all $x:B$, $c(\mathrm{in}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:B$, $y:B$, $z:A$, $p:s(z) =_B x$, and $q:g(z) =_B y$,
$$\mathrm{apd}_c(\mathrm{glue}(x,y,z,p,q)) \equiv c_\mathrm{glue}(x,y,z,p,q)$$

## Properties

### Relation to W-suspensions

[[W-suspensions]] could be constructed as the [[coequalizer type]] of the left and middle dependent pair projections $\pi_{\sum}^A$ and $\pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)}$ of the [[dependent pair type]] $\sum_{x:A} \sum_{y:A} R(x, y)$, both of which have [[domain]] $\sum_{x:A} \sum_{y:A} R(x, y)$ and [[codomain]] $A$:

$$\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) \coloneqq \mathrm{coeq}_{\sum_{x:A} \sum_{y:A} R(x, y),A}(\pi_{\sum}^A, \pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)})$$

Alternatively, given two [[parallel morphisms|parallel]] functions $s:B \to A$ and $t:B \to A$, the coequalizer of $s$ and $t$ could be defined as the W-suspension generated by the graph defined by the family of dependent sum types $\sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)$

$$\underset{x:A,y:A}{\mathrm{Wsus}} \sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)$$

## Examples

* The [[sum type]] of a type $T$ with itself is the coequalizer of two copies of the unique function $\mathrm{rec}_\emptyset^T$ from the [[empty type]] to $T$.
$$T + T \coloneqq \mathrm{coeq}_{\emptyset, T}(\mathrm{rec}_\emptyset^T, \mathrm{rec}_\emptyset^T)$$
* The [[circle type]] is the coequalizer of any two [[endofunctions]] $f:C \to C$ and $g:C \to C$ on a [[contractible type]] $C$.
$$S^1 \coloneqq \mathrm{coeq}_{C, C}(f, g) \quad \mathrm{given} \quad p:\mathrm{isContr}(C)$$
* The [[interval type]] is the coequalizer of the [[constant functions]] with value $\mathrm{false}$ and $\mathrm{true}$ from any [[contractible type]] $C$ to the [[boolean domain]].
$$\mathbb{I} \coloneqq \mathrm{coeq}_{C, \mathrm{bool}}(\mathrm{const}_{C, \mathrm{bool}}(\mathrm{false}). \mathrm{const}_{C, \mathrm{bool}}(\mathrm{true})) \quad \mathrm{given} \quad p:\mathrm{isContr}(C)$$
* More generally, the [[suspension]] of a type $T$ is the coequalizer of the constant functions with value $\mathrm{false}$ and $\mathrm{true}$ from $T$ to the [[boolean domain]].
$$\Sigma T \coloneqq \mathrm{coeq}_{T, \mathrm{bool}}(\mathrm{const}_{T, \mathrm{bool}}(\mathrm{false}). \mathrm{const}_{T, \mathrm{bool}}(\mathrm{true}))$$
* The [[cyclic group]] of [[order]] $n + 1$ for natural number $n:\mathbb{N}$ is the coequalizer of the constant functions with value $0$ and $n + 1$ from any [[contractible type]] $C$ to the [[natural numbers type]].
$$\mathbb{Z}_{n + 1} \coloneqq \mathrm{coeq}_{C, \mathbb{N}}(\mathrm{const}_{C, \mathbb{N}}(0), \mathrm{const}_{C, \mathbb{N}}(n + 1)) \quad \mathrm{given} \quad p:\mathrm{isContr}(C)$$
* The [[pushout]] of a [[span]] $f:C \to A$ and $g:C \to B$ is the coequalizer of the composite functions $\mathrm{in}_{A + B}^A \circ f$ and $\mathrm{in}_{A + B}^B \circ g$, each with domain $C$ and codomain the [[sum type]] $A + B$:
$$A \sqcup_{f,g}^C B \coloneqq \mathrm{coeq}_{C, A + B}(\mathrm{in}_{A + B}^A \circ f, \mathrm{in}_{A + B}^B \circ g)$$

### Propositional truncation

The [[propositional truncation]] of a [[h-set]] $A$ is the [[set truncation]] of the coequalizer of the two product projection functions $\pi_1:A \times A \to A$ and $\pi_2:A \times A \to A$ of the product type $A \times A$

$$\Vert A \Vert_{-1} \coloneqq \Vert \mathrm{coeq}_{A \times A,A}(\pi_1, \pi_2) \Vert_0$$

## See also

* [[coequalizer]], [[homotopy coequalizer]]
* [[propositional truncation]]
* [[W-suspension]]

## References

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

[[!redirects coequalizer type]]
[[!redirects coequalizer types]]

[[!redirects coequaliser type]]
[[!redirects coequaliser types]]


coercion

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A question that is traditionally of some debate in [[type theory]] is,

> What does it mean for a [[term]] to have more than one [[type]]?

This question is relevant in a number of situations:

* the interpretation of [[polymorphism]]

* [[subtyping]]

* [[judgmental equality]] of [[types]]

## Subtyping

One place where the above question is relevant is for type systems with a non-trivial [[subtyping]] relation, in which case it becomes pertinent to consider the meaning of the _subsumption_ rule:

$$\frac{e:A \qquad A \le B}{e:B}$$

One possibility (sometimes called the **inclusion interpretation** of subtyping) is based on the idea of interpreting types as _properties_ of terms (what is also sometimes called "types à la Curry", or the [[intrinsic and extrinsic views of typing|extrinsic]] view of typing).  Under this interpretation, the subsumption rule can be read more or less literally, as asserting that if $e$ satisfies the property $A$, and $A$ entails $B$, then $e$ also satisfies the property $B$.  One potential drawback of this interpretation, though, is that it seems to commit you to giving a meaning to "untyped" terms (although these terms may of course intrinsically have some more coarse-grained type structure, as in models of pure [[lambda calculus]] by [[reflexive object]]s).

A different possibility (sometimes called the **coercion interpretation** of subtyping) is to read the subsumption rule as introducing an implicit _coercion_ from $A$ to $B$.  Under this interpretation, subtyping can be seen as simply providing a convenient shorthand notation, when having to write out these coercions explicitly would be too burdensome.  This interpretation allows one to maintain an [[intrinsic and extrinsic views of typing|intrinsic]] ("à la Church") view of typing, but on the other hand it introduces a subtle issue of _coherence_: if there is more than one way of deriving the subtyping judgment $A \le B$, we would like to know that all of these result in equivalent coercions from $A$ to $B$, so that the meaning of a well-typed term is independent of its typing derivation.

### In dependent type theory

Consider a [[dependent type|dependent]] [[proposition]] $A: Type, x:A \vdash P(x):Prop$. Then the [[dependent sum]], $\sum_{x:A} P(x)$, may be thought of as '$A$s which are $P$'. Sometimes it is convenient to identify a term of this dependent sum as though it were a term of the full type $A$. Here we can use the first projection of the dependent sum to **coerce** the identification.

For example, we might have introduced 'Eve' as a term of type $Woman$, and then wish to form a type $Relatives(Eve)$, when $Relatives(x)$ had been introduced as a type depending on $x: Human$. If $Woman$ had been defined as $\sum_{x:Human} Female(x)$, then $Eve$ has projections to the underlying human and to a warrant for Eve's being female. It would be clumsy to insist on $Relatives(\pi_1(Eve))$. Coercion along this projection allows us to form $Relatives(Eve)$.

[Luo (1999)](#Luo99) proposed a way of formalizing coercions in dependent type theory, through what he calls the _coercive definition rule_, which when generalized to allow dependency on the coerced term is:

$$
\frac{\Gamma, x: B \vdash f(x) : C(x) \; \; \; \Gamma \vdash a : A \;\;\;\Gamma \vdash A \lt_{c} B : Type}{\Gamma \vdash f(a) = f(c(a)) : C(c(a))}
$$

In this notation, a coercion between two types $A$ and $B$ is denoted $A \lt_{c} B$, for $c: (A \to B)$.

### In HoTT

In [[homotopy type theory]], the inclusion of a smaller [[universe|universe in type theory]] in a larger one is often considered as a coercion.

### Coherence

[Reynolds (2000)](#Reynolds00) gave an interesting proof of coherence for a coercion interpretation of a language with subtyping.  The proof works by first building a [[logical relation]] between this "intrinsic semantics" and a separate untyped semantics of the language (the latter based on interpreting programs as elements of a universal [[domain theory|domain]] $U$), and then establishing a "bracketing theorem" which says that for any domain $[[A]]$ in the image of the intrinsic semantics, there are a pair of functions $\phi_A : [[A]] \to U$ and $\psi_A : U \to [[A]]$ such that

* for any $a\in [[A]]$, $a$ is logically related to $\phi_A(a)$
* if $a$ is logically related to $x$, then $a = \psi_A(x)$

Combined, the logical relations theorem and the bracketing theorem imply that the coercion $[[\alpha]] : [[A]] \to [[B]]$ associated to any derivation $\alpha$ of a subtyping judgment $A \le B$ can be factored as

$$
\begin{matrix}
 [[A]] &\overset{\phi_A}{\to} & U & \overset{\psi_B}{\to} & [[B]]
\end{matrix}
$$

and hence in particular that any two derivations $\alpha_1,\alpha_2$ of $A \le B$ result in the same coercion $[[\alpha_1]] = [[\alpha_2]] = (\phi_A;\psi_B) : [[A]] \to [[B]]$.

## Judgmental equality of types

Another place where the above question is relevant in any [[dependent type theory]] which uses a separate [[type]] [[judgment]] and [[judgmental equality]] of types: if two types are judgmentally equal, then every term of the first type is also a term of the second type. This is given from the element conversion rule of judgmental equality,

$$\frac{e:A \qquad A \equiv B}{e:B}$$

which is provable from the structural rules of judgmental equality (cf. section 1.1 & exercise 1.1 of [Rijke22](#Rijke22)).

Judgmental equality of types is the symmetric version of subtyping, since subtyping forms a [[preorder]] on types while judgmental equality forms an [[equivalence relation]] on types, which is a [[symmetric preorder]]. Thus, the same interpretations of subtyping also apply to judgmental equality:

One possibility (sometimes called the **inclusion interpretation** of judgmental equality) is based on the idea of interpreting types as _properties_ of terms (what is also sometimes called "types à la Curry", or the [[intrinsic and extrinsic views of typing|extrinsic]] view of typing).  Under this interpretation, the element conversion rule can be read more or less literally, as asserting that if $e$ satisfies the property $A$, and $A$ is logically equivalent to $B$, then $e$ also satisfies the property $B$. One potential drawback of this interpretation, though, is that it seems to commit you to giving a meaning to "untyped" terms (although these terms may of course intrinsically have some more coarse-grained type structure, as in models of pure [[lambda calculus]] by [[reflexive object]]s).

A different possibility (sometimes called the **coercion interpretation** of judgmental equality) is to read the subsumption rule as introducing an implicit _coercion_ from $A$ to $B$.  Under this interpretation, judgmental equality can be seen as simply providing a convenient shorthand notation, when having to write out these coercions explicitly would be too burdensome. This interpretation allows one to maintain an [[intrinsic and extrinsic views of typing|intrinsic]] ("à la Church") view of typing, but on the other hand it introduces a subtle issue of _coherence_: if there is more than one way of deriving the judgmental equality $A \equiv B$, we would like to know that all of these result in equivalent coercions from $A$ to $B$, so that the meaning of a well-typed term is independent of its typing derivation.

## Related concepts

* [[subtype]]

* [[judgmental equality]]

## References

* {#Reynolds80} [[John C. Reynolds]], _Using Category Theory to Design Implicit Conversions and Generic Operators_, In Proceedings of the Aarhus Workshop on Semantics-Directed Compiler Generation (January 14-18, 1980), Springer-Verlag. ([pdf](http://repository.cmu.edu/cgi/viewcontent.cgi?article=2276&context=compsci))

* {#Reynolds91} John C. Reynolds, _The Coherence of Languages with Intersection Types_. TACS 1991. ([ps](http://www.cs.cmu.edu/afs/cs/user/jcr/ftp/tempdir/coherence.ps))

* {#Reynolds00} John C. Reynolds, _The Meaning of Types: from Intrinsic to Extrinsic Semantics_. BRICS Report RS-00-32, Aarhus University, December 2000. ([pdf](http://repository.cmu.edu/cgi/viewcontent.cgi?article=2290&context=compsci))

* {#Luo99} [[Zhaohui Luo]], Coercive subtyping. Journal of Logic and Computation, 9(1):105–130, 1999. [ps file](http://www.cs.rhul.ac.uk/~zhaohui/JLC98.ps).

* [[Zhaohui Luo]], Type-Theoretical Semantics with Coercive Subtyping, [pdf](http://www.cs.rhul.ac.uk/home/zhaohui/SALT20.pdf)

* {#Rijke22} [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))


coevaluation map

If $(\mathcal{C}, \otimes)$ is a [[monoidal category]] and $X \in \mathcal{C}$ a [[dualizable object]] with [[dual object]] $X^\ast$, the structure morphism

$$
  \mathbb{I} \to X \otimes X^\ast
$$

is also called the _coevaluation map_, in contrast to the other structure map

$$
  X^\ast \otimes X \to \mathbb{I}
$$

which is the [[evaluation map]].

[[!redirects coevaluation maps]]

[[!redirects coevaluation morphism]]
[[!redirects coevaluation morphisms]]

[[!redirects co-evaluation map]]
[[!redirects co-evaluation maps]]

[[!redirects co-evaluation morphism]]
[[!redirects co-evaluation morphisms]]

[[!redirects coevaluation]]
[[!redirects coevaluations]]

[[!redirects co-evaluation]]
[[!redirects co-evaluations]]


coexponential map

The coexponential or symmetrization map is the unique linear map from the symmetric Hopf algebra $S(L)$ on the (underlying vector space of a) Lie algebra $L$ to the enveloping Hopf algebra $U(L)$ of $L$

$$
e : S(L)\to U(L)
$$

such that $e(y)^n = e(y^n)$ for every "linear" element $y\in L\hookrightarrow S(L)$. This map is also given by the formula

$$
y_1\cdots y_k \mapsto \frac{1}{k!} \sum_{\sigma\in\Sigma(k)}\hat{y}_{\sigma(1)}\cdots \hat{y}_{\sigma(n)}
$$

where $y_1,\ldots, y_k\in L\hookrightarrow S(L)$ are arbitrary and
$\hat{y}_1,\ldots,\hat{y}_k$ are the same elements understood in $L\hookrightarrow U(L)$. The coexponential map is an isomorphism of coalgebras $S(L)\cong U(L)$.

The term symmetrization map is more widely used but it is a bit imprecise as it is used also for more general coalgebra isomorphisms $S(L)\to U(L)$ sometimes called also generalized symmetrization maps.

[[!redirects symmetrization map]]


coexponential object

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### Context
#### Category theory
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# Contents
* table of contents
{:toc}

## Idea

The notion of *coexponential objects* is [[abstract duality|dual]] to that of *[[exponential objects]]*.

## Definition

Let $X$ and $Y$ be objects of a [[category]] $C$ such that all binary [[coproducts]] with $Y$ exist.  (Usually, $C$ actually has all binary coproducts.)  Then an __coexponential object__ is an object $Coexp(Y, X)$ equipped with a coevaluation map $\eta \colon X \to Coexp(Y, X) \coprod Y$ which is universal in the sense that, given any object $Z$ and map $e\colon X \to Z \coprod Y$, there exists a unique map $u\colon Coexp(Y, X) \to Z$ such that $e = (u, id_Y) \circ \eta$.

## See also

* [[cocartesian coclosed category]]

* [[locally cocartesian coclosed category]]

* [[exponential object]]

[[!redirects coexponential objects]]


coextension of scalars

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Coextension of scalars is the [[right adjoint]] to _[[restriction of scalars]]_.  It is the dual notion to _[[extension of scalars]]_.

## Definition

Let $f \colon R \to S$ be a [[homomorphism]] of [[algebra|algebraic]] objects such as [[rings]].  If $\cdot_S$ is the left [[action]] of $S$ on some [[module]] $M$, then
$$
  r\cdot_R m \coloneqq f( r )\cdot_S m
$$
defines a left action of $R$ on $M$. This construction extends to a  [[functor]] called *restriction of scalars* along $f$ and denoted
$$
  f^\ast \colon SMod \longrightarrow RMod
$$
This functor has both a [[left adjoint]] and a [[right adjoint]].  The left adjoint $f_!$ is called *[[extension of scalars]]*,  which its right adjoint is called *coextension of scalars* and denoted
$$
  f_* \colon
  RMod
   \longrightarrow
  SMod
$$
Concretely, for any left $R$-module $M$, $f_*(M)$ is the [[hom set]] $RMod(S,M)$ made into a left $S$-module by
$$
    (s g)(s') \coloneqq g(s s')
$$
for $g \in RMod(S,M)$ and $s,s' \in S$.  Here $S$ is made into a left $R$-module by
$$
   r \cdot s \coloneqq f(r) s
$$

## Example

Here is one special case.

For $R$ a [[ring]], write $R$[[Mod]] for its category of [[modules]]. Write [[Ab]] = $\mathbb{Z}$[[Mod]] for the category of [[abelian groups]].

Write $U\colon R Mod \to Ab$ for the [[forgetful functor]] that forgets the $R$-module structure on a module $N$ and just remembers the underlying abelian group $U(N)$.

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

The [[functor]] $U\colon R Mod \to Ab$ has a [[right adjoint]]

$$
  R_* : Ab \to R Mod
$$

given by sending an [[abelian group]] $A$ to the abelian group

$$
  U(R_*(A)) \coloneqq Ab(U(R),A)
$$

equipped with the $R$-[[module]] struture by which for $r \in R$ an element $(U(R) \stackrel{f}{\to} A) \in U(R_*(A))$ is sent to the element $r f$ given by

$$
  r f : r' \mapsto f(r' \cdot r)
  \,.
$$

This is called the **[[coextension of scalars]]** along the ring homomorphism $\mathbb{Z} \to R$.

The [[unit of an adjunction|unit]] of the $(U \dashv R_*)$ [[adjunction]]

$$
  \epsilon_N : N \to R_*(U(N))
$$

is the $R$-module homomorphism

$$
  \epsilon_N : N \to Hom_{Ab}(U(R), U(N))
$$

given on $n \in N$ by

$$
  j(n) : r \mapsto r n
  \,.
$$

=--

## Properties

### Relation to extension of scalars

In some cases coextension of scalars is naturally isomorphic to [[extension of scalars]], which is the *left* adjoint to restriction of scalars.  For example if $H$ is a [[subgroup]] of [[finite number|finite]] [[index of a subgroup|index]] of a group $G$, for any field $k$ there is an inclusion of [[group algebras]]
$$
 i \colon k[H] \to k[G]
$$
and coextension of scalars along $i$ is naturally isomorphic to extension of scalars along $i$.

### Frobenius extensions

Co-Extension of scalars coincides with *[[extension of scalars]]* (to make an [[ambidextrous adjunction]] with [[restriction of scalars]]) in the case of [[Frobenius extensions]], see there for more.

## Related concepts

* [[extension of scalars]] $\dashv$ [[restriction of scalars]] $\dashv$ **coextension of scalars**

## References

* H. Fausk, P. Hu, [[Peter May]], _Isomorphisms between left and right adjoints_ ([pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf))

[[!redirects coextensions of scalars]]


cofiber

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

For $f : A \to B$ a [[morphism]] in a [[category]] with a [[terminal object]], the **cofiber** of $f$ is the [[cofiber coproduct]] of $f$ with the terminal morphism, hence the [[pushout]]

$$
  \array{
    A &\stackrel{f}{\to}& B
    \\
    \downarrow && \downarrow
    \\
    * &\to& B \coprod_A *
  }
  \,.
$$

=--

## Examples

### Cokernels

In an [[additive category]] cofibers over the [[zero object]] are called _[[cokernels]]_.

## Related concepts

* [[fiber]], [[homotopy fiber]], [[fiber sequence]]

* [[homotopy cofiber]]

* [[kernel]], [[cokernel]]

* [[cofiber type]]

[[!redirects cofibre]]
[[!redirects cofibres]]
[[!redirects cofibers]]


cofiber sequence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A cofiber sequence is the dual notion to a [[fiber sequence]].

## Definition

### Abstractly

For $\mathcal{C}$ an [[(∞,1)-category]] with [[(∞,1)-pushouts]], a sequence of [[morphisms]] $A \stackrel{f}{\to} B \to C$ is a _cofiber sequence_ if there is an [[(∞,1)-pushout]] square of the form

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

in $\mathcal{C}$. We say that $C$ is the [[homotopy cofiber]] of $f$.

### Presentation

Under mild conditions on a [[category with weak equivalences]] presenting $\mathcal{C}$ (such as a [[model category]]), homotopy cofibers are presented by _[[mapping cones]]_.

Specifically for cofiber sequences of [[topological spaces]] see at _[[topological cofiber sequence]]_.

## Examples

In a [[stable (∞,1)-category]], every [[fiber sequence]] is also a cofiber sequence and conversely.

### Non-examples

In the unstable case, most fiber sequences are not cofiber sequences or conversely.  For instance, if $0\to K\to G\to H \to 0$ is a [[short exact sequence]] of [[groups]], then the corresponding maps of [[classifying spaces]] $\mathbf{B}K \to \mathbf{B}G \to \mathbf{B}H$ always form a [[fiber sequence]], but not generally a cofiber sequence.

For a concrete counterexample, consider the short exact squence $0 \to \mathbb{Z}\xrightarrow{2} \mathbb{Z}\to \mathbb{Z}/2 \to 0 $.  Upon taking classifying spaces this becomes $S^1 \to S^1 \to RP^{\infty}$, in which the first map is a double cover whose cofiber is $RP^2$.

## Related concepts

* [[long exact sequence in homology]]

  * [[long exact sequence in generalized homology]]

* [[stable (∞,1)-category]]

[[!redirects cofiber sequences]]

[[!redirects cofibre sequence]]
[[!redirects cofibre sequences]]

[[!redirects homotopy cofiber sequence]]
[[!redirects homotopy cofiber sequences]]
[[!redirects homotopy cofibre sequence]]
[[!redirects homotopy cofibre sequences]]

[[!redirects long homotopy cofiber sequence]]
[[!redirects long homotopy cofiber sequences]]
[[!redirects long homotopy cofibre sequence]]
[[!redirects long homotopy cofibre sequences]]

[[!redirects homotopy cofiber]]
[[!redirects homotopy cofibers]]
[[!redirects homotopy cofibre]]
[[!redirects homotopy cofibres]]

[[!redirects cofibration sequence]]
[[!redirects cofibration sequences]]

[[!redirects homotopy cokernel]]
[[!redirects homotopy cokernels]]

[[!redirects homotopy co-fiber]]
[[!redirects homotopy co-fibre]]
[[!redirects homotopy co-fibres]]
[[!redirects homotopy co-fibers]]
[[!redirects homotopy co-kernel]]
[[!redirects homotopy co-kernels]]


cofiber type

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

\tableofcontents

## Definition

In [[dependent type theory]] with [[identity types]], [[function types]], and [[dependent function types]], given a function $f:A \to B$, the **cofiber type** or **cofibre type** of $f$ is the [[higher inductive type]] generated by

* a term $y:\mathrm{cofib}_{A, B}(f)$

* a function $i:B \to \mathrm{cofib}_{A, B}(f)$

* a dependent function
$$\mathrm{glue}:\prod_{x:A} \mathrm{Id}_{\mathrm{cofib}_{A, B}(f)}(i(f(x)),y)$$

In essence, the cofiber type takes the [[image]] of the function $f:A \to B$ and makes it contractible, but leaves the rest of the [[codomain]] untouched.

### Induction rules

Formation rules for cofiber types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash \mathrm{cofib}_{A, B}(f)}$$

Introduction rules for cofiber types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash *:\mathrm{cofib}_{A, B}(f)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash i:B \to \mathrm{cofib}_{A, B}(f)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash \mathcal{g}:\prod_{x:A} * =_{\mathrm{cofib}_{A, B}(f)} i(f(x))}$$

Elimination rules for cofiber types:

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\
\Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x))
\end{array}}{\Gamma \vdash \mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g}):\prod_{y:\mathrm{cofib}_{A, B}(f)} C(y)}$$

Computation rules for cofiber types:

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\
\Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x))
\end{array}}{\Gamma \vdash \beta_{\mathrm{cofib}_{A, B}(f)}^{*}(c_*, c_i, c_\mathcal{g}):\mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g})(*) =_{C(*)} c_*}$$

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\
\Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x))
\end{array}}{\Gamma \vdash \beta_{\mathrm{cofib}_{A, B}(f)}^{i}(c_*, c_i, c_\mathcal{g}):\prod_{y:B} \mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g})(i(y)) =_{C(i(y))} c_i(y)}$$

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\
\Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x))
\end{array}}{\Gamma \vdash \beta_{\mathrm{cofib}_{A, B}(f)}^{\mathcal{g}}(c_*, c_i, c_\mathcal{g}):\prod_{x:A} \mathrm{apd}_{y:\mathrm{cofib}_{A, B}(f).C(y)}(\mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g}), *, i(f(x)), \mathcal{g}(x)) =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_\mathcal{g}(x)}$$

Uniqueness rules for cofiber types:

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\
\Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{y:\mathrm{cofib}_{A, B}(f)} C(y) \quad \Gamma \vdash p:\mathrm{cofib}_{A, B}(f)
\end{array}}{\Gamma \vdash \eta_{\mathrm{cofib}_{A, B}(f)}(c, p):\mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g})(p) =_{C(y)} c(p)}$$

## Examples

* The cofiber type of the unique function out of the empty type to any type $B$ is the type $B + 1$.

* The cofiber type of a function out of the [[unit type]] to any type $B$ is equivalent to $B$.

* The cofiber type of the [[identity function]] on a type $B$ is the [[cone type]] of $B$.

## See also

* [[cofiber]], [[homotopy cofiber]]
* [[pushout type]]
* [[fiber type]]
* [[cone type]]

## References

* *Homotopy Type Theory: Univalent Foundations of Mathematics*,
The Univalent Foundations Program, Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

[[!redirects cofiber type]]
[[!redirects cofiber types]]

[[!redirects cofibre type]]
[[!redirects cofibre types]]


cofibrantly generated model category

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

#Contents#
* table of contents
{:toc}

## Idea  ##

A [[model category]] $C$ is _cofibrantly generated_ if there is a [[set]]
(meaning: [[small set]], not a [[proper class]]) of [[cofibrations]] as well as one of [[acyclic cofibrations]], such that all other cofibrations
and acyclic cofibrations, respectively,
are _generated_ from these, in some sense.

## Definition ##

We need the following general terminology:

\begin{definition}\label{CellsAndInjectives}
**(cells and injectives)**
\linebreak
Let $C$ be a [[category]] with all [[colimits]] and let $S \subset Mor(C)$ a [[class]] of [[morphisms]]. We write:

* $rlp(S)$  for the collection of morphisms with the _right_
  [[lifting property]];

* $llp(S)$ for the collection of morphisms with the _left_  [[lifting property]] with respect to $S$.

Moreover, we also write, now for $I \subset Mor(C)$:

* $cell(I)$ for the [[relative cell complexes]], the class of morphisms obtained by [[transfinite composition]] of [[pushouts]] of [[coproducts]] of elements in $I$;

* $cof(I)$ for the class of [[retracts]] (in the [[arrow category]] $Arr(C)$) of elements in $cell(I)$;

* $inj(I) \coloneqq rlp(I)$ for the class of morphisms with the [[right lifting property]] with respect to $I$, the _$I$-[[injective morphisms]]_.

\end{definition}

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

A [[model category]] $\mathcal{C}$ is **cofibrantly generated** if there are [[small set|small]] [[sets]] of morphisms $I, J \subset Mor(\mathcal{C})$ such that

* $cof(I)$ is precisely the collection of cofibrations of $C$;

* $cof(J)$ is precisely the collection of acyclic cofibrations in $C$; and

* $I$ and $J$ permit the [[small object argument]].

=--

Since $I$ and $J$ are assumed to admit the [[small object argument]] the collection of cofibrations and acyclic cofibrations has the following simpler characterization:

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

In a cofibrantly generated model category (Def. \ref{CofibrantlyGeneratedModelCategory}) we have

* $cof(I) = llp\big(rlp(I)\big)$,

* $cof(J) = llp\big(rlp(J)\big)$.

=--

And therefore the fibrations form precisely $rlp(J)$ and the acyclic fibrations precisely $rlp(I)$.

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

The argument is the same for $I$ and $J$. So take $I$.

By definition we have $I \subset llp\big(rlp(I)\big)$ and it is readily checked that collections of morphisms given by a [[left lifting property]] are stable under [[pushouts]], [[transfinite composition]] and [[retracts]] (see [here](injective+or+projective+morphism#ClosurePropertiesOfInjectiveAndProjectiveMorphisms) for details). So $cof(I) \subset llp\big(rlp(I)\big)$.

For the converse inclusion we use the [[small object argument]]: let $f \colon X \to Z$ be in $llp\big(rlp(I)\big)$. The small object argument produces a factorization
$f : X \stackrel{f' \in cof(I)}{\to} Y \stackrel{f''\in rlp(I)}{\to} Z$.

Finally we apply the "[[retract argument]]": It follows that $f$ has the left [[lifting property]] with respect to $f''$ which yields a morphism $\sigma$ in

$$
  \array{
    X
    &\overset{f'}{\longrightarrow}& Y
    \\
    \mathllap{{}^{f}}
    \big\downarrow
    &{}^\sigma\nearrow&
    \big\downarrow
    \mathrlap{{}^{f''}}
    \\
    Z
    &\underset{=}{\longrightarrow}&
    Z
  }
$$

which exhibits $f$ as a [[retract]] of $f'$

$$
  \array{
    X
    &\overset{=}{\longrightarrow}&
    X
    &\overset{=}{\longrightarrow}&
    X
    \\
    \big\downarrow
    {\mathrlap{{}^f}}
    &&
    \big\downarrow
    {\mathrlap{{}^{f'}}}
    &&
    \big\downarrow
     {\mathrlap{{}^f}}
     \\
     Z
     &\underset{\sigma}{\longrightarrow}&
     Y
     &\underset{f''}{\longrightarrow}&
     Z
     \,.
  }
$$

Therefore $f \in cof(I)$.

=--

## Properties

### Recognition theorem
 {#RecognitionTheorems}

The following theorem allows one to recognize cofibrantly generated model categories by checking fewer conditions.

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

Let $C$ be a [[category]] with all small [[limit]]s and [[colimit]]s and  $W$ a class of maps satisfying [[category with weak equivalences|2-out-of-3]].

If $I$ and $J$ are sets of maps in $C$ such that

1. both $I$ and $J$ permit the [[small object argument]];

1. $cof(J) \subset cof(I) \cap W$;

1. $inj(I) \subset inj(J) \cap W$;

1. one of the following holds

   1. $cof(I) \cap W \subset cof(J)$

   1. $inj(J) \cap W \subset inj(I)$

then there is the stucture of a cofibrantly generated model category
on $C$ with

* weak equivalences $W_C := W$

* generating cofibrations $I$ (i.e. $cof_C := llp(rlp(I))$)

* generating acyclic cofibrations $J$.

=--

This is originally due to [[Daniel Kan]], reproduced for instance as ([Hirschhorn 03, theorem 11.3.1](#Hirschhorn03)).

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

We have to show that with weak equivalences $W$
setting $cof_C := cof(I)$ and $fib_C := inj(J)$ defines a model category
structure.

The existence of [[limit]]s, [[colimit]]s and the [[category with weak equivalences|2-out-of-3 property]] holds by assumption. Closure under retracts of the weak equivalences will hold automatically if we check the rest of the axioms without using it, by an argument of A. Joyal.
Closure under retracts of $fib$ and $cof$ follows by the general statement that
classes of morphisms defined by a left or right lifting property are closed under retracts (e.g. [Hirschhorn 03, 7.2.8, ](#Hirschhorn03)).

The factorization property follows by applying the [[small object argument]]
to the set $I$, showing that every morphism may be factored as
$$
  \stackrel{\in cof(I)}{\to} \stackrel{\in inj(I) \subset fib_C}{\to}
$$
and assumption 3 says that $inj(I) \subset W$. Similarly applying
the small object argument to $J$ gives factorizations
$$
  \stackrel{\in cof(J)}{\to} \stackrel{\in inj(J) = fib_C}{\to}
$$
and assumption 2 guarantees that $cof(J) \subset W$.

It remains to verify the lifting axiom. This verification depends on which
of the two parts of item 4 is satisfied. Assume the first one is, the argument
for the second one is analogous.

Then using the assumption $cof(I) \cap W \subset cof(J)$ and remembering that
we have set $inj(J) = fib_C$ we immediately have the lifting of trivial cofibrations
on the left against fibrations on the right.

To get the lifting of cofibrations on the left with acyclic fibrations on the right,
we show finally that $inj(J) \cap W \subset inj(I)$. To see this, apply the
factorization established before to an acyclic fibration $f : X \to Y$ to get

$$
  \array{
    X &\stackrel{=}{\to}& X
    \\
    \downarrow^{\mathrlap{\in cof(I) \cap W}} && \downarrow^{\mathrlap{f \in inj(J) \cap W}}
    \\
    Z &\stackrel{inj(I)}{\to}& Y
  }
  \,.
$$

With assumption 4 a this is

$$
  \array{
    X &\stackrel{=}{\to}& X
    \\
    \downarrow^{\mathrlap{\in cof(J) }} &&
    \downarrow^{\mathrlap{f \in inj(J) \cap W}}
    \\
    Z &\stackrel{inj(I)}{\to}& Y
  }
$$

so that we have a lift

$$
  \array{
    X &\stackrel{=}{\to}& X
    \\
    {}^{\mathllap{\in cof(J) }}
    \downarrow &{}^\sigma\nearrow&
    \downarrow^{\mathrlap{f \in inj(J) \cap W}}
    \\
    Z &\stackrel{inj(I)}{\to}& Y
  }
$$

which establishes a retract

$$
  \array{
    X &\to& Z &\stackrel{\sigma}{\to}& X
    \\
    \downarrow^f && \downarrow^{\mathrlap{\in inj(I)}} && \downarrow^f
    \\
    Y &\stackrel{=}{\to}& Y
    & \stackrel{=}{\to} & Y
  }
$$

that exhibits $f$ as an element of $inj(I)$ as this is closed under retracts.

=--

### Transfer along adjunctions

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

Let $\mathcal{C}$ be a cofibrantly generated model category, def. \ref{CofibrantlyGeneratedModelCategory}, with generating (acyclic) cofibrations $I$ (and $J$). Let $\mathcal{D}$ be any [[category]] with all small [[limits]] and [[colimits]] and consider a pair of  [[adjoint functors]]

$$
  (F \dashv U)
   \;\colon\;
  \array{
    \mathcal{D}
      \stackrel{\overset{F}{\longleftarrow}}{\underset{U}{\longrightarrow}}
    \mathcal{C}
  }
  \,.
$$

Write $F I \coloneqq \{F(i) | i \in I\}$ and $F J \coloneqq \{F(j) | j \in J\}$. If

1. both $F I$ and $F J$ admit the [[small object argument]];

1. $U$ takes $F J$-[[relative cell complexes]] to weak equivalences

then $F I$, $F J$ induce a cofibrantly generated model structure, def. \ref{CofibrantlyGeneratedModelCategory}, on $\mathcal{D}$. Its weak equivalences are the morphisms that are taken to weak equivalences by $U$. Moreover, the above [[adjunction]] is a [[Quillen adjunction]] for these model structures.

=--

This is due to [[Daniel Kan]], reproduced in ([Hirschhorn 03, theorem 11.3.2](#Hirschhorn03)). See also at _[[transferred model structure]]_.

### Presentation and generation
 {#PresentationAndGeneration}

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

Let $C$ be a cofibrantly generated model category which is also [[proper model category|left proper]]. Then there exists a [[small set]] $S \subset Obj(C)$ of cofibrant objects which detect weak equivalences:

a morphism $f : A \to B$ in $C$ is a weak equivalence, precisely if for all $s \in S$ the induced morphism of [[derived hom-spaces]]

$$
  \mathbb{R}Hom(s,f) : \mathbb{R}Hom(s,A) \to \mathbb{R}Hom(s,B)
$$

is a weak equivalence.

=--

This appears as ([Dugger, prop. A.5](#Dugger)).

## Examples ##

### On simplicial sets

The [classical model category structure on simplicial sets](https://ncatlab.org/nlab/show/model+structure+on+simplicial+sets) is cofibrantly generated:

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

$sSet_{Quillen}$ is a cofibrantly generated model category with

* generating cofibrations the [[boundary]] inclusions $\partial \Delta[n] \to \Delta[n]$;

* generating acyclic cofibrations the [[horn]] inclusions $\Lambda^i[n] \to \Delta[n]$.

=--

### On topological spaces

The category of (based, compactly generated) topological spaces has a cofibrantly generated model structure (the [[classical model structure on pointed topological spaces]]) in which the set of cells is

$$
   I = \{S^{n-1}_+ \rightarrow D^n_+\}_{n\geq 0}
$$

and the set of acyclic cells is

$$
  J = \{D^n_+ \rightarrow (D^n \times I)_+\}_{n\geq 0}
  \,.
$$

(Here $+$ means disjoint basepoint, not northern hemisphere.)

The category of unbased spaces has a similar cofibrantly generated model structure. (The [[classical model structure on topological spaces]].)

### On sequential spectra

The category of [[sequential prespectra]] has two cofibrantly generated model structures (the [[Bousfield-Friedlander model structures]]).

Let $F_d A$ denote a prespectrum whose $n$th space is $\Sigma^{n-d} A$ when $n \geq d$, and $*$ otherwise. Then the _level model structure_ is generated by cells

$$
  I = \{F_d S^{n-1}_+ \rightarrow F_d D^n_+\}_{d,n\geq 0}
$$

([[free spectra]] formed on the standard cells of the [[classical model structure on pointed topological spaces]])

and the set of acyclic cells is

$$
  J = \{F_d D^n_+ \rightarrow F_d (D^n \times I)_+\}_{d,n\geq 0}
$$

The _stable model structure_ has the same cells, but more acyclic cells, which in turn guarantees that the fibrant spectra are the $\Omega$-spectra.

### On structured spectra

The categories of [[symmetric spectra]] and [[orthogonal spectra]] have similar cofibrantly generated level and stable model structures ([[model structure on symmetric spectra]], [[model structure on orthogonal spectra]]).

(see Mandell, May, Schwede, Shipley: _[[Model categories of diagram spectra]]_.)

### On diagrams

The category of diagrams indexed by a fixed small category $D$, taking values in another cofibrantly generated model category $C$.

(The [[model structure on functors]].)

## Related concepts ##

* A cofibrantly generated model category that is also a [[locally presentable category]] is called a *[[combinatorial model category]]*.

* A cofibrantly generated model category for which the [[domains]] of the morphisms in $I$ and $J$ are [[compact objects]] or [[small objects]] is a *[[cellular model category]]*.

* [[cellular category]]

* [[transferred model structure]]

## References ##

The original reference is:

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

In particular, the Kan recognition theorem is in §II.8 and the [[Kan transfer theorem]] is in §II.9.
This manuscript draft later transformed in
[[Homotopy Limit Functors on Model Categories and Homotopical Categories]], losing the content on cofibrantly generated model categories in the process.

Textbook accounts:

* {#Hirschhorn03} [[Philip Hirschhorn]], Sec. 11 in: _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

Further review:

* {#GoerssSchemmerhorn06} [[Paul Goerss]], [[Kirsten Schemmerhorn]], Section 3.1 of: _Model categories and simplicial methods_, Notes from lectures given at the University of Chicago in August 2004, published as: _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics **436** AMS (2007) [[arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436)]

For the general case a useful reference is for instance the first section of

* [[Tibor Beke]], _Sheafifiable homotopy model categories_,  Math. Proc. Cambridge Philos. Soc.  __129__  (2000),  no. 3, 447--475.([math.CT/0102087](http://arxiv.org/abs/math/0102087)), _Sheafifiable homotopy model categories. II_, J. Pure Appl. Algebra  __164__  (2001),  no. 3, 307--324.

For the case of a [[presentable category]]:

* [[Jacob Lurie]], section A.1.2 in: *[[Higher Topos Theory]]*

See also:

* {#Dugger} [[Dan Dugger]], Appendix of: _Replacing model categories with simplicial ones_ ([pdf](http://hopf.math.purdue.edu/Dugger/smod.pdf))

[[!redirects cofibrantly generated model categories]]

[[!redirects cofibrantly generated model structure]]
[[!redirects cofibrantly generated model structures]]

[[!redirects generating cofibration]]
[[!redirects generating cofibrations]]

[[!redirects generating acyclic cofibration]]
[[!redirects generating acyclic cofibrations]]

[[!redirects cofibrantly generated]]

[[!redirects Kan recognition theorem]]


cofibration

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#### Category theory
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#Contents#
* table of contents
{:toc}

## Idea

The notion of _cofibration_ is [[duality|dual]] to that of _[[fibration]]_. See there for more details.

A _cofibration_ is a member of a distinguished *class of cofibrations* in one of the several setups in [[homotopy theory]]:

* [[model category|Quillen categories]] of models for homotopy theory

* the [[category with cofibrations|categories with cofibrations]] of Baues

* [[Waldhausen categories]]

In traditional [[topology]], one usually means a [[Hurewicz cofibration]].

In [[category theory]] and [[descent theory]], Grothendieck however introduced a notion of cofibered category or cofibration, whose definition is categorically dual to that of a [[fibered category]] and based on the generalization of the universal property of coCartesian squares; however it does not correspond to the extension property in topology, but to a lifting property, like for [[fibration]]s. For that reason, Gray suggested (and this is to some extent adopted in $n$lab) to call such categories [[opfibration]]s. In the [[quasicategory|quasicategorical]] setup, the generalizations of fibered categories are called [[Cartesian fibration]]s, and the generalizations of op/co-fibered categories are called coCartesian fibrations.
In the $1$-categorical setup, the coCartesian arrows in op/co-fibrations are indeed the ones which complete coCartesian squares in the special and most important case of [[domain opfibration]]s.

## Examples
 {#Examples}

* [[relative cell complex]] inclusions are the cofibrations in the [[classical model structure on topological spaces]]

* closed [[Hurewicz cofibration]] are the cofibrations in  in the [[Strøm's model category]] on  [[Top]]

* [[monomorphisms]] are the cofibrations in [[Cisinski model structures]] such as the [[classical model structure on simplicial sets]]

## Stability properties

Cofibrations are _usually_ defined in such a way that they are stable at least under the following operations in the category under consideration

 * composition
 * pushouts of spans at least one of whose legs is a cofibration

(Please mind the precise definitions of the category you are using.
Also compare the stability properties of the dual notion [[fibration]].)

## Related concepts

* [[fibration]]
* [[resolution]]
* [[cofibrant object]]
* [[codiscrete cofibration]]

[[!redirects cofibration]]
[[!redirects cofibrations]]
[[!redirects Cofibrations]]


cofibration category

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#### Homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea

A _cofibration category_ is a [[category]] equipped with a sub-[[class]] of its [[morphisms]] which behave like [[cofibrations]].

The notion of _cofibration category_ was introduced by [[Hans-Joachim Baues]] (see references below) as a variant of the category of cofibrant objects, (for which, see [[category of fibrant objects]] and dualise). The axioms are substantially weaker than [[model category|those of  a Quillen model category]], but add one axiom to [[category of fibrant objects|those of K. S. Brown]].

 In the first chapter of his book *Algebraic Homotopy* (see below), [[Hans-Joachim Baues|Baues]] suggests two criteria for an axiom system:

1. _The axioms should be sufficiently strong to permit the basic constructions of homotopy theory._

1. _The axioms should be as weak (and as simple) as possible, so that the constructions of homotopy theory are available in as many contexts as possible._

Baues also introduces the notion of a [[I-category]] being a category with a natural cylinder functor satisfying a set of good properties.

One should distinguish cofibration categories from the Waldhausen's notion of a "category with cofibrations and weak equivalences", nowdays called [[Waldhausen category]].

## Definition

A *cofibration category* is a category $C$ with two classes of morphisms, $\mathit{cof}$ of
_cofibrations_ and $w.e.$ of _weak equivalences_.
These are to satisfy:

C 1) Isomorphisms are both cofibrations and weak equivalences.  If

$$A\stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C$$
are composable morphisms in $C$, then if two of $f$, $g$, $g \circ f$ are in  $w.e.$, so is the
third; if $f$, $g \in \mathit{cof}$ then $g \circ f\in \mathit{cof}$.

C2) Pushout axiom:

Given $B\rightarrow A$ in $\mathit{cof}$ and any $f : B\rightarrow Y$, the pushout

$$
  \array{
    B &\stackrel{f}{\rightarrow}& Y
    \\
    \downarrow_i
    &&
    \downarrow^{\bar{i}}
    \\
   A &\stackrel{\bar{f}}{\to}&  A \cup_B Y
  }
$$

exists and $\bar{i}$ is in $\mathit{cof}$; if $f$ is in $w.e.$, so is $\bar{f}$; if $i$ is also in $w.e.$, so is $\bar{i}$.

C3) Factorisation axiom:

Given $B\stackrel{f}{\rightarrow} Y$, there is a factorisation

$$
  \array{
B&\stackrel{f}{\longrightarrow}&Y\\
\stackrel{i}{\searrow}&&\stackrel{g}{\nearrow}\\
&A&
}$$

with $i \in \mathit{cof}$, $g\in w.e.$

C4) Axiom on fibrant models:

Using the terminology: "$f$ is a _trivial cofibration_" to mean "$f\in
\mathit{cof} \cap w.e.$", and "$RX$ is _fibrant_" to mean "Given any trivial
cofibration $i: RX \stackrel{\simeq}{\rightarrow}Q$, there is a retraction $r :
Q \rightarrow RX$, $r \circ i = Id_{RX}$", the axiom states:

Given $X\in C$, there is trivial cofibration $X\rightarrow RX$ with $RX$ fibrant.

## Examples

* If $C$ has a model category structure then the full subcategory of  cofibrant objects forms a cofibration category.

## Related concepts

* [[fibration category]]

* [[category of cofibrant objects]]

## References

* [[H. J. Baues]], _Algebraic homotopy_, Cambridge studies in advanced mathematics 15,  Cambridge University Press, (1989).

* [[Karol Szumiło]], Homotopy theory of cofibration categories. Homology Homotopy Appl. 18 (2016), no. 2, 345–357.
[doi](https://doi.org/10.4310/HHA.2016.v18.n2.a19)

* [[Karol Szumiło]], Homotopy theory of cocomplete quasicategories. Algebr. Geom. Topol. 17 (2017), no. 2, 765–791.
[doi](https://doi.org/10.2140/agt.2017.17.765)

* [[Krzysztof Kapulkin]], [[Karol Szumiło]], Quasicategories of frames of cofibration categories. Appl. Categ. Structures 25 (2017), no. 3, 323–347.
[doi](https://doi.org/10.1007/s10485-015-9422-y)

* [[Karol Szumiło]], Frames in cofibration categories. J. Homotopy Relat. Struct. 12 (2017), no. 3, 577–616.
[doi](https://doi.org/10.1007/s40062-016-0139-x)

* [[Markus Land]], [[Thomas Nikolaus]], [[Karol Szumiło]], Localization of cofibration categories and groupoid C∗-algebras. Algebr. Geom. Topol. 17 (2017), no. 5, 3007–3020.
[doi](https://doi.org/10.2140/agt.2017.17.3007)

[[!redirects cofibration categories]]

[[!redirects category with cofibrations]]


cofiltered category

\tableofcontents

\section{Idea}

The notion of a cofiltered category is dual to that of a [[filtered category]]. We refer to the latter page for the general theory; here we simply spell out a few things explicitly.

\section{Details}

\begin{defn} A category $\mathcal{C}$ is _cofiltered_ if its [[opposite category]] $\mathcal{C}^{op}$ is [[filtered category|filtered]]. \end{defn}

\begin{rmk} In other words, a cofiltered category is one in which every finite [[diagram]] in $\mathcal{C}$ has a [[cone]]. \end{rmk}

\begin{rmk} \label{RemarkExplicitDefinition} Explicitly, a cofiltered category $\mathcal{C}$ is one for which the following hold.

1. $\mathcal{C}$ has at least one object.
1. For every pair of objects $c_{1}$ and $c_{2}$ of $\mathcal{C}$, there is an object $c_{3}$ of $\mathcal{C}$ such that there exists an arrow $c_{3} \rightarrow c_{1}$ and there exists an arrow $c_{3} \rightarrow c_{2}$.
1. For every pair of objects $c_{1}$ and $c_{2}$ of $\mathcal{C}$, and every pair of arrows $f, g: c_{1} \rightarrow c_{2}$, there is an arrow $h: c \rightarrow c_{1}$ such that $f \circ h = g \circ h$.

\end{rmk}

\begin{rmk} In the final condition of Remark \ref{RemarkExplicitDefinition}, note that $h$ is not required to satisfy any uniqueness condition with regard to the stated property. In particular, it is not necessarily an equaliser of $f$ and $g$. \end{rmk}

\section{Related concepts}

* [[pro-object]]

[[!redirects cofiltered categories]]
[[!redirects cofiltrant category]]
[[!redirects cofiltrant categories]]
[[!redirects cofiltered diagram]]
[[!redirects cofiltered diagrams]]
[[!redirects co-filtered category]]
[[!redirects co-filtered categories]]
[[!redirects co-filtrant category]]
[[!redirects co-filtrant categories]]
[[!redirects co-filtered diagram]]
[[!redirects co-filtered diagrams]]


cofinal

See (or write):

* [[cofinal diagrams]],
* [[cofinal functor]],
* [[cofinal (infinity,1)-functor]],
* [[cofinal subcategory]],
* [[cofinal subset]],
* [[cofinality]],
* [[cofinally small category]].

category: adjective

[[!redirects cofinal]]
[[!redirects cofinally]]


cofinal diagrams

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# Cofinal diagrams
* table of contents
{: toc}

## Definition

A [[pair]] of [[diagrams]] $D,D'$ in a [[category]] $C$ (e.g.: directed systems, sub-posets, ... ) are **cofinal** if they have *equivalent [[colimits]]*.

In most cases where the word "cofinal" is used, it seems to be the case $D'$ is a subdiagram of $D$ in whatever sense of subdiagram appears suitable.  In that case the cofinality is equivalent to the inclusion [[functor]] being a **[[cofinal functor]]**.

As defined above, no relation is posited between $D$ and $D'$, and so it seems not too much in violation of the [[principle of equivalence]] to define cofinalness[^1] as "a single object is a colimit for both diagrams".

It is *not* said that they are final if they have equivalent *[[limits]]* --- the "co"s are not freely mutable, although the dual situation is doubtless just as interesting.

## Examples

* Let $D' = D F$, and let $L\colon D \times (0 \to 1) \to C$ be an initial cocone over $D$; it seems natural to ask that $L \circ (F \times (0 \to 1))$ be an initial cocone over $D'$.

* Nonempty subsets of a finite [[total order]] are cofinal iff they have the same maximum.

* Every infinite subset of $\omega$ is cofinal with $\omega$, as diagrams in $\omega + 1$.

## Related concepts

* [[cofinal functor]]

* [[homotopy final functor]]

* [[final (∞,1)-functor]]

## References

* [[Paolo Perrone]], [[Walter Tholen]], *Kan extensions are partial colimits*, Applied Categorical Structures, 2022. ([arXiv:2101.04531](https://arxiv.org/abs/2101.04531))

[^1]: [[cofinality|Cofinality]] is an ordinal invariant of [[ordinals]], which doesn't make sense in the present generality, so we need another name for the adjective.

[[!redirects cofinal diagram]]


cofinality

# Cofinality
* table of contents
{: toc}

## Idea

The cofinality of a [[quoset]] (quasi-ordered set) is a measure of the size of the quoset and in particular of the size of its tails.  An important special case is the cofinality of an [[ordinal number]], and there is a related concept of the cofinality of a [[cardinal number]].

## Definitions

We begin with definitions that work even in weak [[foundations of mathematics]].

Given a [[quasi-ordered set]] $Q$, the __cofinality__ of $Q$ is the collection of all [[cardinal numbers]] $\kappa$ such that every [[function]] $f\colon [\kappa] \to Q$ (where $[\kappa]$ is any [[set]] of cardinality $\kappa$) has a (strict) [[upper bound]]: an element $x$ of $Q$ such that, whenever $y$ belongs to the [[image]] of $f$, $y \lt x$.  A priori, this collection $Cf(Q)$ may be a [[proper class]], but it is often a [[set]], indeed always in [[classical mathematics]] (as shown below).  We traditionally write $\kappa \lt cf(Q)$ to mean $\kappa \in Cf(Q)$ (for reasons to be seen below).

The __ordinal cofinality__ of $Q$ is the collection $Ocf(Q)$ of all [[ordinal numbers]] $\alpha$ such that ${|\alpha|} \lt cf(Q)$.  This collection is clearly a [[down-set]] and so may be identified with an ordinal number $\ocf(Q)$, also called the __ordinal cofinality__; so we may write $\alpha \lt ocf(Q)$ in place of $\alpha \in Ocf(Q)$, although traditionally we simply write $\alpha \lt cf(Q)$.

If we start with a collection $C$ of [[cardinal numbers]], the __cardinal cofinality__ of $C$ is the collection $Ccf(C)$ of all cardinal numbers $\kappa$ such that, given any $[\kappa]$-indexed [[family]] $F$ of [[sets]], each of which has cardinality in $C$, the [[disjoint union]] of this family (or equivalently the [[union]] in a [[material set theory]]) also has cardinality in $C$.  Again we write $\kappa \lt ccf(C)$ or even $\kappa \lt cf(C)$ to mean $\kappa \in Ccf(C)$.

## Identifications

Assume the [[axiom of choice]].  Then we may identify and simplify some of the concepts above.

*  As a class of cardinal numbers, $cf(Q)$ is clearly a [[down-set]] (that is closed under [[subsets]]), so it must be the set $\{\kappa \;|\; \kappa \lt cf(Q)\}$ for some cardinal number $cf(Q)$, also called the __cofinality__.  (Note that $cf(Q) \leq {|Q|}$, equivalently ${|Q|} \nless cf(Q)$, since the [[identity function]] $Q \to Q$ has no upper bound, so in particular we are *not* dealing with [[proper classes]].)  In this case, we conclude that there is a function $[cf(Q)] \to Q$ that has no strong upper bound, and that $cf(Q)$ is the smallest cardinal number with this property, which is the usual definition.  Assuming that $Q$ is a [[linear order]], it follows that the [[image]] of some function $[cf(Q)] \to Q$ is [[cofinal subset|cofinal]] in $Q$ (whence the terminology).

*  Using the identification of cardinal numbers with certain [[von Neumann ordinals]], the ordinal cofinality $Ocf(Q)$ or $ocf(Q)$ becomes identified with the classical cofinality $cf(Q)$.  (But note that $Cf(Q)$, the collection of cardinal numbers, is only a subset of $Ocf(Q)$ when we identify cardinals as certain ordinals.)

*  Every cardinal cofinality $Ccf(C)$ is also a down-set of cardinal numbers, hence of the form $\{\kappa \;|\; \kappa \lt ccf(C)\}$ for some cardinal number $ccf(C)$.  Furthermore, if we start with a cardinal number $\lambda$ and define the collection $C_\lambda \coloneqq \{\kappa \;|\; \kappa \lt \lambda\}$ of cardinal numbers, then $cf([\lambda]) = ccf(C_\lambda)$.  If we identify $\lambda$ with a von Neumann ordinal, then we also have $cf([\lambda]) = ocf(\lambda)$, so all notions of cofinality agree.

## Properties

### General

Here is an important theorem on ordinal cofinality, which following our definitions is entirely [[constructive mathematics|constructive]]:
$$ ocf(ocf(Q)) = ocf(Q) .$$
In general, an ordinal number $\alpha$ such that $ocf(\alpha) = \alpha$ is called __regular__, so every ordinal cofinality is regular.  For example, $0$, $1$, and $\omega$ are regular ordinals.

A __[[regular cardinal]]__ may be defined to be a collection of cardinals $C$ such that $Ccf(C) = C$.  Assuming the [[axiom of choice]] and making identifications as above, the regular cardinals and the regular ordinals are the same, except that $2$ is a regular cardinal (but not a regular ordinal).  Also, $\{1\}$ is a regular collection of cardinals that is not a down-set, although every other regular cardinal is (and so can be identified with a cardinal number), classically.

Traditionally, one requires a regular ordinal or cardinal to be [[infinite set|infinite]], and thus classically they are the same with no exceptions.

### Cofinality without choice
 {#CofinalityWithoutChoice}

Assuming the consistency of the existence
of arbitrarily large [[strongly compact cardinals]],
it is consistent with Zermelo-Fraenkel [[set theory]] without the [[axiom of choice]] that
every infinite set is a countable union of sets of smaller cardinality.
See [Gitik 1980](#Gitik80).

## Related concepts

* [[initial functor]], [[final functor]]

## References

* {#Gitik80} M. Gitik, _All uncountable cardinals can be singular_, Israel Journal of Mathematics 35:1-2 (1980), 61-88 ([doi:10.1007/bf02760939](http://dx.doi.org/10.1007/bf02760939)).

[[!redirects cofinality]]
[[!redirects cofinalities]]

[[!redirects ordinal cofinality]]
[[!redirects ordinal cofinalities]]

[[!redirects cardinal cofinality]]
[[!redirects cardinal cofinalities]]

[[!redirects confinality]]


cofinally small category

#Definition#

A [[category]] $C$ is **cofinally small** if there is a [[small category]] $C_0$ and a [[cofinal functor]] $C_0 \to C$.


cofinite subset

#Contents#
* table of contents
{:toc}

There are two notions: cofinite subset of a set and a cofinite subset of a (pre)ordered set.

## Definition

Given a [[set]] $X$, then a [[subset]] $S \subset X$ is called _cofinite_ if the [[complement]] $X \backslash S$ is a [[finite set]].

A [[preordered set]] $(\Lambda,\leq)$ is _cofinite_ or closure finite if for every
$\lambda\in\Lambda$ the set of all predecessors of $\lambda$ is finite.

## Related concepts

* [[cofinite topology]]

[[!redirects cofinite subsets]]

[[!redirects cofinite set]]
[[!redirects cofinite sets]]


cofinite topology

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## Definition

Given a [[set]] $X$, then the _cofinite topology_ or _finite complement topology_ on $X$ is the [[topological space|topology]] whose [[open subsets]] are precisely

1. all [[cofinite subsets]];

1. the [[empty set]].

## Properties

If $X$ is a [[finite set]], then its cofinite topology coincides with its [[discrete topology]].

The cofinite topology on a set $X$ is the [[coarse topology|coarsest topology]] on $X$ that satisfies the $T_1$ [[separation axiom]], hence the condition that every [[singleton]] subset is a [[closed subspace]].

Indeed, every $T_1$-topology on $X$ has to be [[fine topology|finer]] that the cofinite topology.

If $X$ is not [[finite set|finite]], then its cofinite topology is not [[sober topological space|sober]], hence in particular not [[Hausdorff topological space|Hausdorff]] (since [[Hausdorff implies sober]]).

A set equipped with the cofinite topology forms a [[compact space]]. However, this type of compact space is not [[uniform space|uniformizable]]; if it were, then under the $T_1$ condition it would also be Hausdorff, which as we saw is not the case.

## References

* Wikipedia, _[Cofiniteness -- Cofinite topology](https://en.wikipedia.org/wiki/Cofiniteness#Cofinite_topology)_

[[!redirects cofinite topologies]]

[[!redirects finite complement topology]]
[[!redirects finite complement topologies]]


coframe

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# Contents
* table of contents
{: toc}

## Idea

The notion of _coframe_ is a generalization of the notion of [[category of closed subsets]] of a [[topological space]]. A _coframe_ is like a category of closed subsets in a space possibly more general than a topological space: a [[locale]]. This in turn is effectively defined to be anything that has a collection of closed subsets that behaves essentially like the closed subsets of a topological space do.

It is also the [[opposite poset]] of a [[frame]].

## Definition

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

A **coframe** $\mathcal{C}$ is

* a [[partial order|poset]]

* that has

  * all [[small limit|small]] [[products]]s, called [[meet|meets]] $\bigwedge$

  * all [[finite colimits]], called [[join|joins]] $\vee$

* and which satisfies the _infinite distributive law_:

  $$
    \bigwedge_i (x\vee y_i) \leq x \vee (\bigwedge_i y_i)
  $$

  for all $x, \{y_i\}_i$ in $A$

  (Note that the converse holds in any case, so we have equality.)

A _coframe homomorphism_ is a homomorphism of posets that preserves finite joins and arbitrary meets. Coframes and coframe homomorphisms form the category Cofrm.

=--

## See also

* [[co-Heyting algebra]]

* [[frame]]

* [[cotopos]]

## References

* Mamuka Jibladze, "Cosheaves, coframes, cotoposes: some new facts, some old questions" ([web archive](https://web.archive.org/web/20130602214748/http://www.wra1th.plus.com/gcw/rants/math/PSSL/1997.html))