enriched category

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#### Enriched category theory
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## Idea

The notion of _enriched category_ is a generalization of the notion of [[category]].

Very often instead of merely having a _[[set]]_ of [[morphism]]s from one [[object]] to another, a category will have a _[[vector space]]_ of morphisms, or a _[[topological space]]_ of morphisms, or some other such thing.  This suggests that we should take the definition of ([[locally small category|locally small]]) category and generalize it by replacing the [[hom-set]]s by _hom-objects_, which are objects in a suitable category $K$.
This gives the concept of 'enriched category'.

The category $K$ must be [[monoidal category|monoidal]], so that we can define composition as a morphism
$$ \circ : hom(y,z) \otimes hom(x,y) \to hom(x,z)  $$
So, a __category enriched over $K$__ (also called a __category enriched in $K$__, or simply a __$K$-category__), say $C$, has a collection $ob(C)$ of objects and for each pair $x,y \in ob(C)$, a '[[hom-object]]'
$$ hom(x,y) \in K .$$
We then mimic the usual definition of category.

We may similarly define a _functor enriched over $K$_
and a _natural transformation enriched over $K$_, obtaining a [[strict 2-category]] of $K$-enriched categories, [[category of V-enriched categories|$K$-Cat]].  By general 2-category theory, we thereby obtain notions of _$K$-enriched adjunction_, _$K$-enriched equivalence_, and so on.

There is also an enriched notion of [[limit]] called a [[weighted limit]], but it is somewhat more subtle (and in particular, it is difficult to construct purely on the basis of the 2-category $K$-Cat).

More generally, we may allow $K$ to be a [[multicategory]], a [[bicategory]], a [[double category]], or a [[virtual double category]].

See also [[enriched category theory]].

## Definition

Ordinarily enriched categories have been considered as _enriched over_ or _enriched in_ a [[monoidal category]]. This is discussed in the section

* [Enrichment in a monoidal category](#InMonoidCat)

More generally, one may think of a monoidal category as a [[bicategory]] with a single object (so that the monoidal product becomes [[horizontal composition]]) and this way regard enrichment in a monoidal category as the special case of _enrichment in a bicategory_ . This is discussed in the section

* [Enrichment in a bicategory](#InBicat)

Enriched categories and [[enriched functors]] between them form themselves a category, the _[[category of V-enriched categories]]_.

### Enrichment in a monoidal category {#InMonoidCat}

Let $V$ be a [[monoidal category]] with

* tensor product $\otimes : V \times V \to V$;

* tensor unit $I \in Obj(V)$;

* associator $\alpha_{a,b,c} : (a \otimes b)\otimes c \to a \otimes (b \otimes c)$;

* left unitor $l_a : I \otimes a \to a$;

* right unitor $r_a : a \otimes I \to a$.

A (small) **$V$-category** $C$ (or **$V$-enriched category** or **category enriched over/in $V$**) is

* a [[set]] $Obj(C)$ -- called the set of objects;

* for each ordered pair $(a,b) \in Obj(C) \times Obj(C)$ of objects in $C$ an object $C(a,b) \in Obj(V)$ -- called the [[hom-object]] or _object of morphisms_ from $a$ to $b$;

* for each ordered triple $(a,b,c)$ of objects of $C$ a morphism $\circ_{a,b,c} : C(b,c) \otimes C(a,b) \to C(a,c)$ in $V$ -- called the _composition morphism_;

* for each object $a \in Obj(C)$ a morphism $j_a : I \to C(a,a)$ -- called the _identity [[global element|element]]_

* such the following diagrams commute:

for all $a,b,c,d \in Obj(C)$:

$$
  \array{
     (C(c,d)\otimes C(b,c)) \otimes C(a,b)
     &&\stackrel{\alpha}{\to}&&
     C(c,d) \otimes (C(b,c) \otimes C(a,b))
     \\
     \downarrow^{\circ_{b,c,d}\otimes Id_{C(a,b)}}
     &&&& \downarrow^{Id_{C(c,d)}\otimes \circ_{a,b,c}}
     \\
     C(b,d)\otimes C(a,b)
     &\stackrel{\circ_{a,b,d}}{\to}&
       C(a,d)
     &\stackrel{\circ_{a,c,d}}{\leftarrow}&
     C(c,d) \otimes C(a,c)
  }
$$

this says that composition in $C$ is _associative_;

and

$$
  \array{
    C(b,b)\otimes C(a,b)
    &\stackrel{\circ_{a,b,b}}{\to}&
    C(a,b)
    &\stackrel{\circ_{a,a,b}}{\leftarrow}&
    C(a,b) \otimes C(a,a)
    \\
    \uparrow^{j_b \otimes Id_{C(a,b)}}
    & \nearrow_{l}&& {}_r\nwarrow&
    \uparrow^{Id_{C(a,b)}\otimes j_a}
    \\
    I \otimes C(a,b)
    &&&&
    C(a,b) \otimes I
  }
$$

this says that composition is _unital_.

A monoidal category over which one intends to do enriched category theory may be referred to as a _base of enrichment_; this applies also to [enrichment in a bicategory](#InBicat).

In practice, one often takes a (monoidal) base of enrichment to be a [[Bénabou cosmos]], in order to have available all the infrastructure needed to carry out common categorical constructions such as enriched [[functor categories]], [[tensor products of enriched categories]], enriched [[ends]] and [[coends]], enriched [[limits]] and [[colimits]], and so on.

#### Enrichment through lax monoidal functors ####

If $V$ is a monoidal category, then an alternative way of viewing a $V$-category is as a set $X$ together with a (lax) [[monoidal functor]] $\Phi = \Phi_d$ of the form

$$V^{op} \stackrel{yon_V}{\to} Set^{V} \stackrel{d}{\to} Set^{X \times X}$$

where the codomain is identified with the monoidal category of spans on $X$, i.e., the local hom-category $\hom(X, X)$ in the [[bicategory]] of [[span]]s of sets. Given an $V$-category $(X, d: X \times X \to V)$ under the ordinary definition, the corresponding monoidal functor $\Phi$ takes an object $v$ of $V^{op}$ to the span

$$\Phi(v)_{x, y} := \hom_V(v, d(x, y))$$

Under the composition law, we get a natural map

$$\hom(v, d(x, y)) \times \hom(v', d(y, z)) \to \hom(v \otimes v', d(x, y) \otimes d(y, z)) \stackrel{\hom(1, comp)}{\to} \hom(v \otimes v', d(x, z))$$

which gives the tensorial constraint $\Phi(v) \circ \Phi(v') \to \Phi(v \otimes v')$ for a monoidal functor; the identity law similarly gives the unit constraint.

Conversely, by using a Yoneda-style argument, such a monoidal functor structure on $\Phi = \Phi_d$ induces an $M$-enrichment on $X$, and the two notions are equivalent.

Alternatively, we can equivalently describe a $V$-enriched category as precisely a bicontinuous lax monoidal functor of the form

$$Set^V \to Set^{X \times X}$$

since bicontinuous functors of the form $Set^V \to Set^{X \times X}$ are precisely those of the form $Set^d$ for some function $d: X \times X \to V$, at least if $V$ is [[Cauchy complete category|Cauchy complete]].

### Enrichment in a bicategory
 {#InBicat}

Let $B$ be a [[bicategory]], and write $\otimes$ for horizontal (1-cell) composition (written in Leibniz order). A [[category enriched in a bicategory|category enriched in the bicategory]] $B$ consists of a [[set]] $X$ together with

* A function $p: X \to B_0$,
* A function $\hom: X \times X \to B_1$, satisfying the typing constraint $\hom(x, y): p(x) \to p(y)$,
* A function $\circ: X \times X \times X \to B_2$, satisfying the constraint $\circ_{x, y, z}: \hom(y, z) \otimes \hom(x, y) \to \hom(x, z)$,
* A function $j: X \to B_2$, satisfying the constraint $j_x: 1_{p(x)} \to \hom(x, x)$,

such that the associativity and unitality diagrams, as written above, commute. If $M$ is a monoidal category, we can view it as a 1-object bicategory by working with its [[delooping#delooping_of_higher_structures|delooping]] $\mathbf{B} M$; the notion of enrichment in $M$ coincides with the notion of enrichment in the bicategory $\mathbf{B} M$.

If $X$, $Y$ are sets which come equipped with enrichments in $B$, then a $B$-functor consists of a function $f: X \to Y$ such that $p_Y \circ f = p_X$, together with a function $f_1: X \times X \to B_2$, satisfying the constraint $f_1(x, y): \hom_X(x, y) \to \hom_Y(f(x), f(y))$, and satisfying equations expressing coherence with the composition and unit data $\circ$, $j$ of $X$ and $Y$. (Diagram to be inserted, perhaps.)

Categories enriched in bicategories were originally introduced by [[Bénabou]] under the name **polyad**. These can be seen as [[horizontal categorification|many-object generalisations]] of [[monads]], since a category $X$ enriched in a bicategory $B$ is precisely a monad in $B$ when $X$ has one object.

### Enrichment in a double category {#InDoubleCat}

It is also natural to generalize further to categories enriched in a (possibly weak) [[double category]].  Just like for a bicategory, if $D$ is a double category, then a $D$-**enriched category** $\mathbf{X}$ consists of a set $X$ together with

* for each $x\in X$, an object $p(x)$ of $D$,
* for each $x,y\in X$, a horizontal arrow $\hom(x, y)\colon p(x) \to p(y)$ in $D$,
* for each $x,y,z\in X$, a 2-cell in $D$:
  $$\array{p(x) & \overset{hom(x,y)}{\to} & p(y) & \overset{hom(y,z)}{\to} & p(z) \\
  \Vert && \circ_{x,y,z} && \Vert\\
  p(x) && \underset{hom(x,z)}{\to} && p(z)}$$
* for each $x\in X$, a 2-cell in $D$:
  $$\array{p(x) & \overset{id}{\to} & p(x)\\
    \Vert && \Vert\\
    p(x) & \underset{hom(x,x)}{\to} & p(x)}$$

satisfying analogues of the associativity and unit conditions.  Note that is is exactly the same as a category enriched in the horizontal bicategory of $D$; the vertical arrows of $D$ play no role in the definition.  However, they *do* play a role when it comes to define functors between $D$-enriched categories.  Namely, if $\mathbf{X}$ and $\mathbf{Y}$ are $D$-enriched categories, then a $D$-functor $f\colon \mathbf{X}\to \mathbf{Y}$ consists of:

* a function $f\colon X\to Y$,
* for each $x\in X$ a vertical arrow $f_x\colon p_X(x) \to p_Y(f(x))$ in $D$,
* for each $x,y\in X$ a 2-cell in $D$:
  $$\array{p(x) & \overset{hom_X(x,y)}{\to} & p(y)\\
    ^{f_x}\downarrow && \downarrow^{f_y}\\
    p(f(x))& \underset{hom_Y(f(x),f(y))}{\to} & p(f(y))}$$

satisfying suitable equations.  If $D$ is vertically discrete, i.e. just a bicategory $B$ with no nonidentity vertical arrows, then this is just the same as a $B$-functor as defined above.  However, for many $D$ this notion of functor is more general and natural.

### Other bases of enrichment

It is possible to generalise the above further to enrichment in [[virtual double categories]] (see [Leinster (2002)](#Leinster02)), which generalises enrichment in a [[multicategory]].

Other kinds of enrichment:

- Enrichment in a [[closed category]] (though this is subsumed by enrichment in a (closed) multicategory).
- Enrichment in a [[skew-monoidal category]].

## Change of enriching category {#BaseChange}

Also called "change of base", this typically refers to a [[monoidal functor]] between bases of enrichment $V \to W$, inducing a [[2-functor]] $V$-$Cat \to W$-Cat, enabling constructions in $V$-$Cat$ to be transferred to constructions in $W$-Cat, as follows.

### Passage between ordinary categories and enriched categories

Every $K$-enriched category $C$ has an [[underlying ordinary category]], usually denoted $C_0$, defined by $C_0(x,y) = K(I, hom(x,y))$ where $I$ is the unit object of $K$.

If $K(I, -): K \to Set$ has a left adjoint $- \cdot I: Set \to K$ (taking a set $S$ to the tensor, aka the [[power|copower]] $S \cdot I$, viz. the coproduct of an $S$-indexed set of copies of $I$), then any ordinary category $C$ can be regarded as enriched in $K$ by forming the composite

$$Ob(C) \times Ob(C) \stackrel{\hom}{\to} Set \stackrel{-\cdot I}{\to} K$$

These two operations form [[adjoint functors]] relating the [[2-category]] [[Cat]] to the 2-category $K$-Cat.

### Lax monoidal functors

More generally, any (lax) [[monoidal functor]] $F: K \to L$ between monoidal categories can be regarded as a "[[change of enriching category|change of base]]".  By applying $F$ to its hom-objects, any category enriched over $K$ gives rise to one enriched over $L$, and this forms a [[2-functor]] from $K$-Cat to $L$-Cat, and in fact from $K$-Prof to $L$-Prof; see [[profunctor]] and [[2-category equipped with proarrows]].

Moreover, this operation is itself functorial from $MonCat$ to $2Cat$.  In particular, any [[monoidal adjunction]] $K\rightleftarrows L$ gives rise to a [[2-adjunction]] $K Cat\rightleftarrows L Cat$ (and also for profunctors).  The adjunction $Cat \rightleftarrows K Cat$ described above is a special case of this arising from the adjunction $-\cdot I: Set \rightleftarrows K : K(I,-)$.

This and further properties of such "[[change of enriching category|change of base]]" are explored in [Crutwell 14](#Crutwell14)

## Tensor product of enriched categories
 {#TensorProductOfEnrichedCategories}

If $V$ is a [[symmetric monoidal category]], then there is a tensor product of **V-enriched categories** (see *[[enriched product category]]*) which makes the category **[[VCat|$V Cat$]]** of V-enriched categories itself a symmetric monoidal category. In fact V-Cat is even a [[symmetric monoidal 2-category]]. See [Kelly (1982), p. 12](#Kelly82).

## Enrichment versus internalization

See [[enrichment versus internalisation]].

## Examples
 {#Examples}

* A category enriched in [[Set]] is a [[locally small category]].

* A category enriched in [[chain complexes]] is a [[dg-category]].

* A category enriched in [[simplicial sets]] is called a [[simplicially enriched category|simplicial category]], and these form one model for [[(∞,1)-categories]].

  Beware: the term 'simplicial category' is also used to mean a category [[internalization|internal to]] simplicial sets.  In fact, a category enriched in simplicial sets is a special case of a category internal to simplicial sets, namely one where the simplicial set of objects is discrete.

* A category enriched in [[Top]] is a [[topologically enriched category]].  These are also a model for [[(∞,1)-categories]].

  Again beware: the term 'topological category' is perhaps more commonly used to mean a category [[internalization|internal to]] Top.  People also use it for _[[topological concrete category]]_. And again: a category enriched in Top is a special case of one internal to Top, namely one where the space of objects is discrete.

* A category enriched in [[Cat]] is a [[strict 2-category]].

* A category enriched in [[Grpd]] is a [[strict (2,1)-category]].

* A [[strict n-category]] is a category enriched over strict $(n-1)$-categories. In the limit $n \to \infty$ this leads to [[strict omega-category|strict omega-categories]].

* An [[algebroid]], or [[linear category]], is a category enriched over [[Vect]].  Here $Vect$ is the category of vector spaces over some fixed [[field]] $K$, equipped with its usual tensor product.  It is common to emphasize the dependence on $K$ and call a category enriched over Vect a **$K$-linear category**.

* More generally, if $K$ is any [[commutative ring]], a category enriched over $K\,$[[Mod]] is sometimes called a $K$-linear category.

* In particular, taking $K$ to be $\mathbb{Z}$ (the ring of [[integers]]), a [[ringoid]] (or [[Ab-enriched category]]) is a category enriched over [[Ab]].

* A (Lawvere) [[metric space]] is a category enriched over the poset $([0, \infty], \geq)$ of extended positive real numbers, where $\otimes$ is $+$.

* An [[ultrametric space]] is a category enriched over the poset $([0, \infty], \geq)$ of extended positive real numbers, where $\otimes$ is $\max$.

* A [[preorder]] is a category enriched over the category of [[truth value]]s, where $\otimes$ is [[logical conjunction|conjunction]].

* An [[apartness relation|apartness space]] is a [[groupoid]] enriched over the opposite of the category of truth values, where $\otimes$ is [[disjunction]].

* A [[torsor]] over some group $G$ may be modeled by a category enriched over the [[discrete category]] on the set $G$, where $\otimes$ is the group operation.  Not every such category determines a torsor, however; it must be nonempty as well as [[Cauchy complete category|Cauchy complete]].

* [[F-categories]] are categories enriched over a subcategory of $Cat^{\to}$, and similarly [[M-categories]] are categories enriched over a subcategory of $Set^{\to}$.

* [[2-categories with contravariance]] (and generalizations such as [[3-categories with contravariance]]) can also be described as enriched categories.

* [[tangent bundle categories]] can be described as a certain kind of enriched category with certain [[powers]]; see [Garner 2018](#Garner2018) for details.

* [[Lawvere theories]] can be represented as enriched categories as well; see [Garner 2013](#Garner2014) and [Garner and Power 2017](#GarnerPower2017) for details.

## Related concepts

* [[Bénabou cosmos]]

* [[enriched poset]]

* [[enriched groupoid]]

* [[enriched presheaf]]

* [[enriched monad]]

* [[enriched bicategory]]

* [[category enriched over a bicategory]]

* [[model structure on enriched categories]]

* [[enriched monoidal category]]

* [[enriched model category]]

* [[cartesian closed enriched category]], [[locally cartesian closed enriched category]]

* [[enriched (∞,1)-category]]

* [[enriched type]]

* [[magnitude of an enriched category]]

## References

The idea that it is worthwhile to replace [[hom-sets]] by "[[hom-objects]]" (in some sense) is perhaps first present in

* [[Saunders MacLane]], §24 of: *Categorical algebra* Bulletin of the American Mathematical Society 71.1 (1965): 40-106.

The earliest accounts of enriched categories were given independently in:

* [[Jean Bénabou]], *Catégories relatives*, C. R. Acad. Sci. Paris **260** (1965) 3824-3827 [[gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4019v/f37.item)]

* [[Jean-Marie Maranda]], *Formal categories*, Canadian Journal of Mathematics **17** (1965) 758-801 [[doi:10.4153/CJM-1965-076-0](https://doi.org/10.4153/CJM-1965-076-0), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7C463460EB8CAC64C2CA340F870CF80/S0008414X00039729a.pdf/formal-categories.pdf)]

(both of which also introduce the notion of *[[strict 2-categories]]* as the example of [[Cat]]-enriched categories)

though Kelly also gave an account for enrichment in arbitrary categories (then deducing necessary structure on a tensor product):

* [[Gregory Maxwell Kelly]]. _Tensor products in categories_. Journal of Algebra **2** 1 (1965) 15-37 [doi:10.1016/0021-8693(65)90022-0]

See also:

* [[Fred Linton]]. _Autonomous categories and duality of functors_. Journal of Algebra 2.3 (1965): 315-349.

Enrichment in a [[multicategory]] was first suggested in:

* {#Lambek69} [[Joachim Lambek]], *Deductive Systems and Categories II: Standard constructions and closed categories*,  In: [[Peter Hilton]] (eds.) *Category Theory, Homology Theory and their Applications I*, Lecture Notes in Mathematics **86** Springer 1969 ([doi:10.1007/BFb0079385](https://doi.org/10.1007/BFb0079385), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0079385.pdf))

Textbook accounts:

* {#Kelly82} [[Max Kelly]], _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

* [[Francis Borceux]], Chapter 6 of: _[[Handbook of Categorical Algebra]] Vol 2: Categories and Structures_, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

* [[Emily Riehl]], _Basic concepts of enriched category theory_, chapter 3 in: _[[Categorical Homotopy Theory]]_, Cambridge University Press, 2014 ([pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf), [doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457))

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 1.3 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

Also:

* [[Francis Borceux]], I. Stubbe, _Short introduction to enriched categories_ ([pdf](http://www-lmpa.univ-littoral.fr/~stubbe/PDF/EnrichedCatsKLUWER.pdf))

With an eye towards application in [[homotopy theory]]:

* {#Richter19} [[Birgit Richter]], Section 9 of: _From categories to homotopy theory_, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 ([doi:10.1017/9781108855891](https://doi.org/10.1017/9781108855891), [book webpage](https://www.math.uni-hamburg.de/home/richter/catbook.html), [pdf](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf))

Discussion of [[change of enriching category]] is in

* {#Crutwell14} [[Geoff Cruttwell]], chapter 4 of _Normed spaces and the Change of Base for Enriched Categories_, 2014 ([pdf](http://pages.cpsc.ucalgary.ca/~gscruttw/publications/thesis4.pdf))

Vista of some modern generalizations is in

* [[Tom Leinster]], _Generalized enrichment for categories and multicategories_,  [math.CT/9901139](http://arxiv.org/abs/math.CT/9901139)

* {#Leinster02} [[Tom Leinster]], _Generalized enrichment of categories_, Journal of Pure and Applied Algebra __168__ (2002), no. 2-3, 391-406, [math.CT/0204279](http://arxiv.org/abs/math.CT/0204279)

* John Armstrong, _[Enriched categories](http://unapologetic.wordpress.com/2007/08/13/enriched-categories/)_

Further examples are discussed in

* {#Garner2013} [[Richard Garner]], *Lawvere theories, finitary monads and Cauchy-completion*, Journal of Pure and Applied Algebra, 2014, [arxiv](https://arxiv.org/abs/1307.2963)

* {#GarnerPower2017} [[Richard Garner]] and [[John Power]], *An enriched view on the extended finitary monad--Lawvere theory correspondence*, 2017
([arXiv:1707.08694
](https://arxiv.org/abs/1707.08694)

* {#Garner2018} [[Richard Garner]], *An embedding theorem for tangent categories*, Advances in Mathematics, 2018, [doi](https://www.sciencedirect.com/science/article/pii/S0001870817303122), [arxiv](https://arxiv.org/abs/1704.08386)

On enrichment in more general structures:

* [[Christian Lair]], _Systèmes tensoriels et systèmes enrichis_, Diagrammes 43 (2000): 3-48.

[[!redirects enriched categories]]
[[!redirects category enriched]]
[[!redirects categories enriched]]
[[!redirects enriched]]
[[!redirects polyad]]


enriched category theory

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#Contents#
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## Idea

**Enriched category theory** is the [[category theory]] of [[enriched categories]].

Pretty much every notion of ordinary category theory has its analog in enriched category theory, and, by taking the enriching category to be [[Set]], enriched category theory subsumes all of the theory of [[locally small categories]].

Enriched categories have a multitude of uses and applications, that makes studying their general theory quite worthwhile.

## Applications

### As a tool in higher category theory

Enriched categories may be used to model objects in [[higher category theory]]: if the objects of the enriching category $V$ behave themselves as [[(n,r)-categories]], then a $V$-[[enriched category]] behaves like a (possibly special) $(n+1,r+1)$-category.

For instance

* a [[Cat]]-enriched category is a [[strict 2-category]];

* since a [[simplicial set]] that is a [[Kan complex]] is a model for an [[∞-groupoid]], a [[Kan complex]]-enriched category is a model for an [[(∞,1)-category]];

* since a simplicial set that is a [[quasi-category]] is a model for an [[(∞,1)-category]], a quasi-category-enriched category is a model for an [[(∞,2)-category]].

In practice it is often useful to handle enriched categories such that only certain enriched [[full subcategories]] of them are the the desired models for objects in higher category theory. For instance often it is useful to work with general [[sSet-categories]] and have a prescription for how to find a [[Kan complex]]-enriched full subcategory inside them.

This is achieved notably by combining enriched category theory with [[model category]] theory:

an [[enriched model category]] or more generally an [[enriched homotopical category]] is an enriched category with extra information on how it behaves as a model in higher category. Notably [[sSet-model categories]] serve as models for [[(∞,1)-category theory]].

##Literature##

Textbook accounts include

* [[Max Kelly]], _Basic Concepts of Enriched Category Theory_, Cambridge University Press, Lecture Notes in Mathematics 64 (1982)

  Republished as: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 ([tac:10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

* [[Francis Borceux]], Vol 2, chapter 6 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

Also

* [[Francis Borceux]], I. Stubbe, _Short introduction to enriched categories_ ([pdf](http://www-lmpa.univ-littoral.fr/~stubbe/PDF/EnrichedCatsKLUWER.pdf))

* [[Emily Riehl]], Chapter 3 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

For more references see at _[[enriched category]]_.

## Entries on Enriched Category Theory

### Monoidal categories

* [[monoidal category]]

  * [[tensor product]]

* [[closed category]]

  * [[internal hom]]

* [[closed monoidal category]]

* [[cartesian monoidal category]]

* [[braided monoidal category]], [[symmetric monoidal category]]

* [[Day convolution]]

* [[Bénabou cosmos]]

### Enriched functors

* [[enriched functor]]

* [[strong functor]]

* [[closed functor]], [[cartesian closed functor]]

### Generalizations of monoidal categories

* [[multicategory]]

* [[bicategory]]

* [[double category]]

* [[virtual double category]]

### Enriched categories

* [[enriched category]]

* [[enriched functor]]

  * [[enriched functor category]]

* [[category of V-enriched categories]]

* [[profunctor]]

* [[change of enriching category]]

### Universal constructions

* [[weighted limit]]

* [[end|ends and coends]]

* [[Kan extension]]

### Homotopical enrichment

* [[enriched homotopical category]]

* [[enriched model category]]

* [[model structure on homotopical presheaves]]

[[!redirects Enriched Category Theory]]
[[!redirects enriched]]


Enriched Category Theory > history

< [[Enriched Category Theory]]
[[!redirects Enriched Category Theory -- history]]


enriched category theory contents

**[[enriched category theory]]**

## Background

* [[category theory]]

* [[monoidal category]], [[closed monoidal category]]

* [[cosmos]], [[multicategory]], [[bicategory]], [[double category]], [[virtual double category]]

## Basic concepts

* [[enriched category]]

* [[enriched functor]], [[profunctor]]

* [[enriched natural transformation]]

* [[enriched adjoint functor]]

* [[enriched product category]]

* [[enriched functor category]]

## Universal constructions

* [[weighted limit]]

* [[end]], [[coend]]

## Extra stuff, structure, property

* [[copowering]] ([[tensoring]])

  [[powering]] ([[cotensoring]])

* [[monoidal enriched category]]

  * [[cartesian closed enriched category]]

  * [[locally cartesian closed enriched category]]

### Homotopical enrichment

* [[enriched homotopical category]]

* [[enriched model category]]

* [[model structure on homotopical presheaves]]

[[!redirects enriched category theory - contents]]


enriched derivator

## Idea

If a [[derivator]] is regarded as a "shadow" of an $(\infty,1)$-category, then an *enriched derivator* is an analogous shadow of an [[enriched category|enriched]] $(\infty,1)$-category.

+--{: .query}
[[Peter LeFanu Lumsdaine]]: What does "enriched $(\infty,1)$-category" mean here? --- just one of the models for $(\infty,1)$-categories as (Top-, Kan-, SSet-)enriched categories, or actually some further idea of "$(\infty,1)$-category enriched in something"?  (The latter sounds an interesting idea, but I'm not quite sure how to imagine it!)

[[Mike Shulman]]: I had the latter in mind.  Just as a 1-category can be enriched over a monoidal 1-category, an $(\infty,1)$-category should be enrichable over a monoidal $(\infty,1)$-category.  The "default" enrichment will be over $\infty$-groupoids (which one could express in any particular model), just as the "default" enrichment for 1-categories is over sets.
=--

## Definitions

There are several different things in which one could try to enrich a derivator.  It would be nice to be able to remain completely in the world of derivators by enriching a derivator over a [[monoidal derivator]].  However, it seems unlikely that the naive notion of monoidal derivator (a [[pseudomonoid]] in the 2-category of derivators) contains enough information to make this feasible, so we either have to augment that notion somehow, or enrich over something else.  At present, the latter is easier.

### Enrichment over a monoidal $(\infty,1)$-category

+--{: .standout}
The following is tentative.
=--

Let $V'$ be a bicomplete closed symmetric [[monoidal (∞,1)-category]], and let $V\subset V'$ be a [[reflective sub-(∞,1)-category]] which is a 1-category and which is closed under the monoidal structure and the [[internal-hom]].  For example:

* $V'=\infty Gpd$ with cartesian [[product]], $V=Set$.
* $V'= \infty Gpd_*$ (pointed $\infty$-groupoids) with [[smash product]], $V=Set_*$.

We can therefore talk about $V$-enriched categories.  Moreover, since $V$ is bicomplete and closed, the category $V$-$Cat$ is again closed symmetric monoidal, and hence enriched over itself.

+--{: .un_defn}
###### Definition
A **$(V,V')$-enriched prederivator** is a $V CAT$-enriched functor
$$ D\colon V Cat^{op} \to V CAT. $$
(or replacing $V Cat$ by some subcategory of it.)
=--

For instance, when $(V,V') = (Set, \infty Gpd)$ then this is precisely the ordinary notion of prederivator.  We define this by reference to $V$, rather than $V'$, in order that the definition of prederivator can remain a purely 1-categorical notion, without referring explicitly to any $(\infty,1)$-category.

We now need to specify the axioms which an enriched prederivator should satisfy to be called an enriched derivator.  Most of them are easy by analogy:

* **(Der1)** $D$ takes [[coproducts]] to [[products]].

* **(Der2)** If $1$ denotes the unit $V$-category, then for any $V$-category $A$ the family of functors $D(A) \to D(1)$ is jointly [[conservative functor|conservative]].

* **(Der3)** For any $V$-functor $u\colon A\to B$, the $V$-functor $u^*\colon D(B) \to D(A)$ has a left and a right adjoint $u_!$ and $u_*$.

* **(Der5)** For any object $x\in V$, let $I[x]$ denote the $x$-fattened [[interval category]], with two objects $0$ and $1$ and $hom(0,1)=x$.  Then for any $V$-category $A$, the induced functor $D(A\times I[x]) \to D(A)^{I[x]}$ is essentially surjective and full.

The last, and most important, axiom, involves a characterization of the appropriate [[homotopy exact squares]] of $V$-categories.  Here is finally where the larger $(\infty,1)$-category $V'$ enters the picture.  If $A$ is a $V$-category and $F\colon A\to V$ and $G\colon A^{op}\to V$ are $V$-functors, then we can take their [[tensor product]] $G\otimes_A F \in V$, but we can also regard them as landing in $V'$ and take their homotopy tensor product $G \otimes^h_A F \in V'$, and these will generally give different results.

* **(Der4)** Given any square
  $$ \array{ A & \overset{f}{\to} & B \\
   ^u\downarrow & & \downarrow ^v \\
   C & \underset{g}{\to} & D} $$
  of $V$-categories such that for any objects $c\in C$ and $b\in B$, the induced map
  $$ C(-,c) \otimes^h_A B(b,-) \to D(v(b),g(c)) $$
  is an equivalence in $V'$, then the induced transformation
  $$ u_! f^* \to g^* v_! $$
  (and hence also its mate $v^* g_* \to f_* u^*$) is an isomorphism.

This should be compared with the characterizations of [[exact squares]] for ordinary (enriched) category theory and of [[homotopy exact squares]].

A $(V,V')$-enriched prederivator satisfying (Der1)--(Der5) is called a **$(V,V')$-enriched derivator**.

### Enrichment over a monoidal homotopical category

Note that all we really needed of $V'$ was a place for a well-behaved homotopy tensor product of $V$-profunctors to live.  Thus, we could replace it by anything else sufficiently good, such as a [[monoidal model category]] or a [[closed monoidal homotopical category]].  In this case, the homotopy tensor product can be computed as a "homotopy coend" or more explicitly as a [[two-sided bar construction]].

Moreover, since the only dependence on $V'$ in the definition is via the homotopy tensor product of functors, the particular choice of model we make is essentially irrelevant for the resulting notion of derivator.  In other words, the true input for a notion of enriched derivator is an ordinary monoidal category $V$, together with "a homotopy theory in which $V$ is reflectively embedded."

## Examples

* When $(V,V') = (Set,\infty Gpd)$, all the axioms of an enriched derivator are identical to those of an ordinary [[derivator]], except for (Der4).  The usual axiom (Der4) asserts the conclusion only for [[comma squares]].  However, Cisinski's theorem characterizing [[homotopy exact squares]] shows that this implies the enriched version of (Der4) stated above (which is a priori stronger).  To make this comparison with the way Cisinski's theorem is usually stated, one has to observe that $C(-,c) \otimes^h_A B(b,-)$ can be constructed as a [[two-sided bar construction]], which in turn can (in this case) be identified with the nerve of a particular category.

* Conjecturally, $(Set_*, \infty Gpd_*)$-enriched derivators are the same as [[pointed derivators]] in the usual sense.

* Also conjecturally, any $V'$-enriched $(\infty,1)$-category should have an underlying $(V,V')$-enriched derivator.

## Properties

Just as an ordinary derivator encodes a well-behaved notion of [[homotopy limits]], an enriched derivator encodes a notion of homotopy [[weighted limit]].  In order to recover this, we need to represent [[profunctors]] by their [[cograph of a profunctor|collages]].

Specifically, let $H\colon A⇸ B$ be a $V$-profunctor, and $A \overset{u}{\to} \bar{H} \overset{v}{\leftarrow} B$ its collage.  Then homotopy $H$-weighted limits in an enriched derivator are computed by the composite $v^* u_*$, and similarly $H$-weighted colimits are computed by $u^* v_!$.

Note that unlike in the ordinary unenriched case, it is not possible to compute all limits by Kan extending along functors to the terminal category.  This is because of the presence and importance of weighted limits.

### Homotopy Kan extensions are pointwise

Part of the intuition behind the usual axiom (Der4) is that it says that the homotopy Kan extensions which make up the structure of a derivator are all [[pointwise Kan extension|pointwise]].  For left Kan extensions along $u\colon A\to B$, pointwiseness means that each object $(Lan_u X)(b)$ is calculated as a suitable colimit, which in the unenriched case can be expressed as an ordinary colimit over the comma category $u/b$ -- hence why we assert that comma squares are exact.

In the enriched situation, the (co)limit involved in the notion of "pointwise Kan extension" is irreducibly a [[weighted limit]], and not constructible as a comma object.  However, the Kan extensions in an enriched derivator are still pointwise in a suitable sense; we can see this as follows.  Let $u\colon A\to B$ be a $V$-functor, let $b\in B$, let $B(u,b)\colon 1 ⇸ A$ denote the representable profunctor, and let $H$ be its collage.  Then the square
$$\array{ A & \to & A \\
\downarrow & & \downarrow^u\\
H & \to & B } $$
satisfies the hypothesis of the enriched (Der4).  (This is just the homotopical version of [[Yoneda reduction]]: $B(u,b) \otimes_A^h A \simeq B(u,b)$.)  Therefore, if we left extend along $u$, then restrict to $H$ and further restrict to $b$, we get the same thing as if we just extended to $H$ and restricted to $b$ --- but the latter is exactly the process of taking the $B(u,b)$-weighted colimit.  Thus, the evaluation of $u_! X$ at $b$ is computed as a weighted colimit, and hence the Kan extension is "pointwise."

[[!redirects enriched derivators]]


enriched directed loop graph

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

# Contents
* table of contents
{: toc}

## Idea ##

A directed loop graph may be understood as a [[thin category|thin]] (0,1)-directed [[pseudograph]], hence as a thin directed pseudograph enriched over the [[cartesian monoidal category|cartesian monoidal]] [[poset]] of [[truth values]].
In generalization, one may speak of enriching directed loop graphs over other [[monoidal posets]].

## Defintion ##

Let $(M, \leq, \wedge, \top)$ be a [[monoidal poset]], such as a [[meet-semilattice]]. A **$M$-enriched directed loop graph** or **directed loop graph enriched over/in $M$** is a [[set]] $P$ with a [[binary function]] $o:P \times P \to M$

## Examples ##

* An ordinary directed loop graph is just an $\Omega$-enriched directed loop graph, with $\Omega$ the set of [[truth values]].

* A [[Lawvere metric space]] is a $\overline{\mathbb{R}_{\geq 0}}$-enriched [[proset]], where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* A [[quasipseudometric space]] is a $\mathbb{R}_{\geq 0}$-enriched [[proset]], where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* A [[pseudometric space]] is a $\mathbb{R}_{\geq 0}$-enriched [[symmetric proset]], where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* A [[metric space]] is a $\mathbb{R}_{\geq 0}$-enriched [[set]], where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

## See also ##

* [[meet-semilattice]]
* [[pseudometric space]]
* [[enriched proset]]
* [[premetric space]]

[[!redirects enriched directed loop graphs]]


enriched factorization system

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An ([[orthogonal factorization system|orthogonal]]) [[factorization system]] in a [[category]] $C$ consists of two classes of [[morphisms]] $(E,M)$ such that every morphism in $C$ factors as an $E$-morphism followed by an $M$-morphism, and we have $e$ satisfies the [[left lifting property]] --uniquely for the orthogonal case, $e\perp m$, see [[orthogonality]] --  for any $e\in E$ and $m\in M$ .

To [[enriched category|enrich]] this, we should first consider enriched orthogonality.  The statement $e\perp m$ for maps $e:A\to B$ and $m:X\to Y$ can be rephrased by saying that the square
$$
\array{C(B,X) & \to & C(A,X)\\
  \downarrow && \downarrow \\
  C(B,Y) & \to & C(A,Y)}
$$
is a [[pullback]] in [[Set]].

More generally, the statement that there exists at least one lift is to say that the canonical morphism

$$
  C(B,X) \to C(A,X) \prod_{C(A,Y)} C(B,Y)
$$

into the [[pullback]] is a [[split epimorphism]].

It is then clear that if $C$ is enriched over some [[monoidal category]] $V$, to say that $e\perp m$ in an enriched sense, we should instead require this square to be a pullback of enriched [[hom-objects]] in $V$.  Note, though, that $e$ and $m$ are still maps in the underlying ordinary category $C_0$ of $C$.  Likewise, the factorization property still only makes sense for maps in $C_0$.

Therefore, we define an **enriched (orthogonal) factorization system** on an enriched category $C$ to consist of two classes of maps $(E,M)$ in $C_0$ such that

1. $e\perp m$ in the above enriched sense for every $e\in E$ and $m\in M$, and
1. Every map in $C_0$ factors as an $E$-map followed by an $M$-map.

By the definition of $C_0$, enriched orthogonality implies ordinary orthogonality.  Therefore, an enriched factorization system on $C$ induces an ordinary factorization system on $C_0$.  Conversely, if $C$ has [[power]]s that preserve the maps in $M$, or [[copower]]s that preserve the maps in $E$, then unenriched orthogonality in $C_0$ implies enriched orthogonality by a Yoneda lemma argument.

## Functoriality

Moreover, the factorization functor can be made into an enriched functor in the following way.  There is a $V$-category $C^{\mathbf{2}}$ whose objects are morphisms in $C_0$ and whose hom-objects are defined by, for $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$, a pullback
$$
\array{C^{\mathbf{2}}(f_1,f_2) & \to & C(X_1,X_2)\\
  \downarrow && \downarrow \\
  C(Y_1,Y_2) & \to & C(X_1,Y_2).}
$$
(This is the [[power]] of $C$ by ${\mathbf{2}}$ in the 2-category $V-Cat$, and also the $V$-functor category $[{\mathbf{2}}_V,C]$, where ${\mathbf{2}}_V$ denotes the free $V$-category on ${\mathbf{2}}$.)

Likewise, we have $C^{\mathbf{3}}$ whose objects are composable pairs $X\overset{f}{\to} Y \overset{g}{\to} Z$ of morphisms in $C_0$, and whose hom-objects are defined by pullbacks
$$
\array{C^{\mathbf{3}}((f_1,g_1),(f_2,g_2)) & \to & C^{\mathbf{2}}(f_1,f_2)\\
  \downarrow && \downarrow \\
  C^{\mathbf{2}}(g_1,g_2) & \to & C(Y_1,Y_2).}
$$
By functoriality we then mean that the factorization is given by a functor $C^{\mathbf{2}} \to C^{\mathbf{3}}$ which, when composed with the "composition" functor $C^{\mathbf{3}} \to C^{\mathbf{2}}$, gives the identity of $C^{\mathbf{2}}$.

The interesting part of the enrichment of this functor is the following: given $f_1:X_1\to Z_1$ and $f_2:X_2\to Z_2$ in $C^{\mathbf{2}}$, with factorizations $X_1 \overset{m_1}{\to} Y_1 \overset{e_1}{\to} Z_1$ and $X_2 \overset{m_2}{\to} Y_2 \overset{e_2}{\to} Z_2$, by enriched orthogonality we have a pullback
$$
\array{C(Y_1,Y_2) & \to & C(X_1,Y_2)\\
  \downarrow & & \downarrow\\
  C(Y_1,Z_2) & \to & C(X_1,Z_2)}
$$
and we also have a commutative square
$$
\array{C^{\mathbf{2}}(f_1,f_2) & \to & C(X_1,X_2) & \to & C(X_1,Y_2)\\
  \downarrow &&&& \\
  C(Z_1,Z_2) &&&& \downarrow\\
  \downarrow &&&& \\
  C(Y_1,Z_2) & & \longrightarrow & & C(X_1,Z_2)
}
$$
inducing a map $C^{\mathbf{2}}(f_1,f_2) \to C(Y_1,Y_2)$.  It is then straightforward to construct the rest of the functor.

This argument, as it depends crucially on the universality of the pullback and hence the uniqueness part of orthogonality, fails for [[weak factorization system]]s.  Although they can be made functorial in many cases, rarely can their functoriality be made enriched (as far as is known).

## References
 {#References}

* [[Emily Riehl]], Chapter 13 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

[[!redirects enriched factorization systems]]


enriched functor

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

#Contents#
* table of contents
{:toc}

## Idea

Enriched functors are used in place of [[functors]] in [[enriched category theory]]: like functors they send objects to objects, but instead of mapping [[hom-sets]] to [[hom-sets]] they assign morphisms in the enriching category between [[hom-objects]], while being compatible with composition and units in the obvious way.

## Definition

Given two categories $C, D$ [[enriched category|enriched in]] a [[monoidal category]] $V$, an **enriched functor** $F: C \to D$ consists of

* A function $F_0: C_0 \to D_0$ between the underlying collections of objects;
* A $(C_0 \times C_0)$-indexed collection of morphisms of $V$,
$$F_{x, y}: C(x, y) \to D(F_0x, F_0y)$$
where $C(x, y)$ denotes the hom-object in $V$,

* such that the following diagrams commute for all $a, b, c \in C_0$:

  * respect for composition:
    $$
      \array{
        C(b,c) \otimes C(a,b)
        &\stackrel{\circ_{a,b,c}}{\to}&
        C(a,c)
        \\
        \downarrow^{F_{b,c} \otimes F_{a,b}} && \downarrow^{F_{a,c}}
        \\
        D(F_0(b), F_0(c)) \otimes D(F_0(a), F_0(b))
        &\stackrel{\circ_{F_0(a),F_0(b), F_0(c)}}{\to}&
        D(F_0(a), F_0(c))
      }
    $$

  * respect for units:
    $$
      \array{
        && I
        \\
        & {}^{j_a}\swarrow && \searrow^{j_{F_0(a)}}
        \\
        C(a,a)
        &&\stackrel{F_{a,a}}{\to}&&
        D(F_0(a), F_0(a))
      }
    $$

     $V$'s unit object $I$ and the family of identity [[generalized element |elements]] $J_i$ from it are defined in [[enriched category]].

## Properties

### General

* Analogous to the [[functor category]] for ordinary [[functors]] between ordinary [[categories]], enriched functors between two enriched categories form an [[enriched functor category]].

### Relation to strong functors

For [[closed monoidal categories]] $V$ there is a close relation between $V$-enriched functors and $V$-[[strong functors]].

For the moment see at *[enriched monad -- relation to strong monads](enriched+monad#RelationToStrongMonads)* for more.

## Examples

* [[linear functor]]

* [[smooth functor]]

* [[topologically enriched functor]]

* Consider the enriching category with $[0,\infty]$ the set of objects, maps given by $\geq$, and tensor by $+$. Then $V$-categories are [[Lawvere metric spaces]], and $V$-functors are distance-decreasing maps.

* Categories enriched in the [[poset]] of truth values with conjunction as monoidal product are posets, and the corresponding enriched functors are precisely order-preserving maps.

## Related concepts

* [[enriched natural transformation]]

* [[enriched functor category]]

* [[enriched (∞,1)-functor]]

## References

* [[Max Kelly]], _Basic Concepts of Enriched Category Theory_ ([web](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html))

*  {#Lawvere73}  [[Bill Lawvere]] (1973).  _Metric spaces, generalized logic and closed categories_.  Reprinted in [[TAC]], 1986.  [Web](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html).

* [[Emily Riehl]], §3.5 in *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

For more references see at _[[enriched category theory]]_.

[[!redirects enriched functors]]


enriched functor category

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of $V$-[[enriched category theory]] (with $V$ any suitable [[cosmos for enrichment]]) for $\mathbf{C}$, $\mathbf{D}$ a [[pair]] of [[enriched categories|$V$-enriched categories]], there is, fist of all, an ordinary [[category]] whose

* [[objects]] are the [[enriched functors]] $F \colon \mathbf{C} \longrightarrow \mathbf{D}$,

* [[morphisms]] $\eta \colon F \Rightarrow F'$ are the [[enriched natural transformations]].

This notion is a generalization of the plain notion of the *[[functor category]]* between [[locally small categories]], to which it reduces in the case that $V \equiv $ [[Set]] equipped with its [[cartesian monoidal category|cartesian monoidal]]-[[structure]].

As such one may and does call this ordinary category the "enriched functor category" between $\mathbf{C}$ and $\mathbf{D}$, in the sense of the "category of enriched functors".

However, this category of enriched functors canonically enhances further to a $V$-[[enriched category]] itself, hence to an "enriched-functor enriched-category", then traditionally often denoted $[\mathbf{C}, \mathbf{D}]$ or similar.

Namely, for
$
  F, G \,\colon\, \mathbf{C} \longrightarrow \mathbf{D}
$
a [[pair]] of $V$-[[enriched functors]] between $V$-[[enriched categories]], the collection of [[enriched natural transformations]] from $F$ to $G$ can also be given the [[structure]] of an [[object]] in $V$, in a compatible way:

## Definition

For $\mathbf{C}$ and $\mathbf{D}$ $V$-[[enriched categories]], there is a $V$-[[enriched category]], denoted $[\mathbf{C},\,\mathbf{D}]$, whose

* [[objects]] are the $V$-[[enriched functors]] $F \colon \mathbf{C} \longrightarrow \mathbf{D}$

* [[hom-objects]] $\;[C,D](F,G)$ between $V$-functors $F, G$ are given by the $V$-enriched [[end]]
$$
  [\mathbf{C},\mathbf{D}](F,G)
  \;\coloneqq\;
  \textstyle{\int}_{c \in \mathbf{C}}
  \mathbf{D}\big(F(c),\, G(c)\big)
$$
over the [[enriched hom-functor]]
$$
  \mathbf{D}\big(
    F(-)
    ,\,
    G(-)
  \big)
  \;\colon\;
  \mathbf{C}^{op}
    \otimes
  \mathbf{C}
    \longrightarrow
  V
  \,,
$$

* the [[composition]] operation

$$
  \circ_{K,F,G}
    \,\colon\,
  [\mathbf{C},\mathbf{D}](F,G)
    \otimes
  [\mathbf{C},\mathbf{D}](K,F)
   \longrightarrow
  [\mathbf{C},\mathbf{D}](K,G)
$$
is the universal morphism into the [[end]] $[\mathbf{C},\mathbf{D}](K,F)$ obtained from observing that the composites
$$
  [\mathbf{C},\mathbf{D}](F,G)
    \otimes
  [\mathbf{C},\mathbf{D}](K,F)
    \stackrel{E_c\otimes E_d}{\longrightarrow}
  \mathbf{D}\big(F(c),G(c)\big)
    \otimes
  \mathbf{D}\big(K(c),F(c)\big)
    \stackrel{\circ_{K(c), F(c), G(c)}}{\longrightarrow}
  \mathbf{D}\big(K(c), F(c)\big)
$$

  (where $E_c \colon [\mathbf{C},\mathbf{D}](F,G) \to \mathbf{D}\big(F(c),G(c)\big)$ denotes the canonical [[morphism]] out of the [[end]], i.e. the counit)

  form an [[extranatural transformation|extra $V$-natural]] family, equivalently that
$$
 [\mathbf{C},\mathbf{D}](F,G)
   \otimes
 [\mathbf{C},\mathbf{D}](K,F)
   \stackrel
    {\prod_{c \in Obj(c)}E_c\otimes E_c}
    {\longrightarrow}
 \prod_{c \in Obj(c)}
 \mathbf{D}\big(F(c),G(c)\big)
   \otimes
 \mathbf{D}\big(K(c),\,F(c)\big)
 \stackrel{\prod_{c \in Obj(c)}
   \circ_{K(c), F(c), G(c)}}{\longrightarrow}
 \prod_{c \in Obj(c)}
   \mathbf{D}\big(K(c),\,G(c)\big)
$$
equalizes the two maps appearing in the [[equalizer]]-definition of the [[end]].

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

For $V = $[[Set]], so that $V$-enriched categories are just ordinary [[locally small category|locally small categories]], the $V$-enriched functor category coincides with the ordinary [[functor category]].

=--

(For more see the example [below](#OrdinaryFunctorCategories).)

## Examples

### Ordinary functor categories
 {#OrdinaryFunctorCategories}

To understand the role of the [[end]] here, it is useful to spell this out for the case where $V =$ [[Set]], where we are dealing with ordinary [[locally small category|locally small]] categories.

So let $V = Set$ where set is equipped with its [[cartesian monoidal category|cartesian monoidal structure]].

Recall the definition of the [[end]] over

$$
  \mathbf{D}\big(
    F(-),G(-)
  \big)
  \,\colon\,
  \mathbf{C}^{op} \otimes \mathbf{C}
    \longrightarrow
  Set
$$

as an [[equalizer]]: it is the [[universal property|universal]] [[subobject]]

$$
  \textstyle{\int}_{c \in C} \mathbf{D}\big(F(c),\, G(c)\big)
  \hookrightarrow
  \prod_{c \in Obj(C)} \mathbf{D}\big(F(c),\, G(c)\big)
$$

of the [[product]] of all [[hom-sets]] in $D$ between the [[images]] of [[objects]] in $C$ under the [[functors]] $F$ and $G$. So one element $ \eta \in \int_{c \in \mathbf{C}} \mathbf{D}\big(F(c), G(c)\big)$ is a collection of morphisms

$$
  \big(
    F(c)
      \stackrel{\eta_c}{\longrightarrow}
    G(c)
  \big)_{c \in Obj(c)}
$$

such that the "left and right [[action]]" (in the sense of [[distributors]]) of $\mathbf{D}\big(F(-),G(-)\big)$ on these elements coincides. Unwrapping what this action is (see the details at [[end]]) one find that

* the "right [[action]]" by a morphism $c \stackrel{f}{\to} d$ is the postcomposition $
  (F(c) \stackrel{\eta_c}{\to} G(c)) \mapsto
  (F(c) \stackrel{\eta_c}{\to} G(c) \stackrel{G(f)}{\to} G(d))
$

* the "left [[action]]" by a morphism $c \stackrel{f}{\to} d$ is the precomposition $
  (F(d) \stackrel{\eta_d}{\to} G(d)) \mapsto
  (F(c) \stackrel{F(f)}{\to} F(d) \stackrel{\eta_d}{\to} G(d) )
$.

So the [[invariants]] under the combined action are those $\eta$ for which for all $f \colon c \to d$ in $C$ the [[diagram]]
$$
  \array{
    F(c) &\stackrel{\eta_c}{\longrightarrow} & G(c)
    \\
    \Big\downarrow\mathrlap{{}^{F(f)}}
    &&
    \Big\downarrow\mathrlap{{}^{G(f)}}
    \\
    F(d)
    &
      \stackrel{\eta_d}{\longrightarrow}
    &
    G(d)
  }
$$

[[commuting diagram|commutes]]. Evidently, this means that the elements $\eta$ of the [[end]] $\int_{c \in \mathbf{C}} \mathbf{D}\big(F(c), G(c)\big)$ are precisely the [[natural transformation]]s between $F$ and $G$.

### Pointwise order

For categories enriched in [[truth values]], the enriched functor category is given by the [[pointwise order]].

## Properties
 {#Properties}

### As internal hom in $V$-Cat
 {#AsInternalHomInVCat}

If the [[cosmos for enrichment|enriching cosmos]] $(V, \otimes)$ is ([[closed monoidal category|closed]], [[complete category|complete]] and) [[symmetric monoidal category|symmetric monoidal]] then forming the $V$-enriched functor category out of any $V$-enriched category $\mathcal{C}$ is [[right adjoint]] to forming the [[enriched product category]] with $\mathcal{C}$, hence serves as the [[internal hom]] in [[VCat|$V Cat$]].

[[Kelly (1982), §2.3](#Kelly82)]

### Enhanced enrichment
 {#EnhancedEnrichment}

It happens that a $V$-enriched functor category $Func(\mathbf{X},\mathbf{C})$ --- which by the above discussion is a priori a $\mathbf{V}$-enriched category --- carries an enhanced enrichment over the functor category $Func(\mathbf{X},\mathbf{V})$.

A common kind of example are categories of [[simplicial objects]] in an ordinary $Set$-enriched category -- hence functor categories $Func(\Delta^{op}, C)$ on the [[opposite category|opposite]] [[simplex category]] -- which one wants to regard not just enriched in $Set$, but as enriched in $Func(\Delta^{op}, Set) \,\equiv\,$ [[sSet]], namely in [[simplicial sets]].

We may bootstrap the discussion of this situation by first considering the case where $\mathbf{C} = \mathbf{V}$ itself, where it means that $Func(\mathbf{X},\mathbf{V})$ is closed monoidal under the $\mathbf{X}$-objectwise tensor product taken in $\mathbf{V}$.
In the base case where $\mathbf{V} = $ [[Set]] this reduces to the statement of the [[closed monoidal structure on presheaves]].

\begin{proposition}
\label{EnhancedEnichmentFormulas}
Let

* $(V, \otimes)$ be a [[bicomplete category|bicomplete]] [[closed monoidal category]], canonically regarded as a $V$-[[enriched category|enriched]] $\mathbf{V}$

  via its [[internal hom]] $[-,-]$;

* $\mathbf{C}$ a [[bicomplete category|bicomplete]] $V$-[[enriched category]] which is

  also $V$-[[tensoring|tensored]], via $\mathbf{V} \times \mathbf{C} \overset{(\text{-})\cdot(\text{-})}{\longrightarrow} \mathbf{C}$,

  and $V$-[[cotensoring|cotensored]] via $\mathbf{V}^{op} \times \mathbf{C} \overset{(\text{-})^{(\text{-})}}{\longrightarrow} \mathbf{C}$;

* $\mathbf{X}$ a [[small category|small]] $V$-enriched category.

Then the enriched functor category...

1. ...$\mathbf{V}^{\mathbf{X}} \,\coloneqq\, Func(\mathbf{X},\,\mathbf{V})$ carries [[closed monoidal category]]  structure

   given by the $\mathbf{X}$-objectwise [[tensor product]]

   $$
     \mathcal{S}_{\mathbf{X}}
     ,\,
     \mathcal{T}_{\mathbf{X}}
     \,\in\,
     \mathbf{V}^{\mathbf{X}}
     \;\;\;\;\;\;\;\;\;\;
     \vdash
     \;\;\;\;\;\;\;\;\;\;
     \mathcal{S}_{\mathbf{X}}
     \otimes
     \mathcal{T}_{\mathbf{X}}
     \;\colon\;
     x \;\mapsto\;
     (\mathcal{S}_{\mathbf{X}} \otimes \mathcal{T}_{\mathbf{X}})_x
     \;\;
       \coloneqq
     \;\;
     \mathcal{S}_x \otimes \mathcal{T}_x
   $$

   and with corresponding [[internal hom]] given by:

   $$
     \mathcal{S}_{\mathbf{X}}
     ,\,
     \mathcal{T}_{\mathbf{X}}
     \,\in\,
     \mathbf{V}^{\mathbf{X}}
     \;\;\;\;\;\;\;\;\;\;
     \vdash
     \;\;\;\;\;\;\;\;\;\;
     \big[
       \mathcal{S}_{\mathbf{X}}
       ,\,
       \mathcal{T}_{\mathbf{X}}
     \big]
     \;\colon\;
     x \;\mapsto\;
     \big[
       \mathcal{S}_{\mathbf{X}}
       ,\,
       \mathcal{T}_{\mathbf{X}}
     \big]_x
     \;\;
       \coloneqq
     \;\;
     \textstyle{\int}_{x' \in \mathbf{X}}
     \mathbf{V}\big(
       \mathbf{X}(x,x')
         \otimes
       \mathcal{S}_{x'}
       ,\,
       \mathcal{T}_{x'}
     \big)
     \,.
   $$

1. ... $\mathbf{C}^{\mathbf{X}}$ becomes [[enriched category|enriched]], [[tensoring|tensored]] and  [[cotensoring|cotensored]] over $\mathbf{V}^{\mathbf{X}}$, via the following [[end]]-formulas, respectively:

   $$
     \begin{array}{rccl}
       \mathscr{V}_{\mathbf{X}}
       ,\,
       \mathscr{W}_{\mathbf{X}}
       \;\in\;
       \mathbf{C}^{\mathbf{X}}
       &\;\;\;\vdash\;\;\;&
       \mathbf{C}^{\mathbf{X}}
       \big(
         \mathscr{V}_{\mathbf{X}}
         ,\,
         \mathscr{W}_{\mathbf{X}}
       \big)
       &
       \coloneqq\;
       \Big(
         x
           \,\mapsto\,
         \textstyle{\int}_{x' \in \mathbf{X}}
         \mathbf{C}\big(
           \mathbf{X}(x,x')
           \cdot
           \mathscr{V}_{x'}
           ,\,
           \mathscr{W}_{x'}
         \big)
       \Big)
       \;\;\;
       \in
       \;
       \mathbf{V}^{\mathbf{X}}
       \\
       \mathcal{S}_{\mathbf{X}}
       \,\in\, \mathbf{V}^{\mathbf{X}}
       ;\;
       \mathscr{V}_{\mathbf{X}}
       \in\,
       \mathbf{C}^{\mathbf{X}}
       &\vdash&
       \mathcal{S}_{\mathbf{X}}
       \cdot
       \mathscr{V}_{\mathbf{X}}
       &
       \coloneqq\;
       \big(
         x
         \,\mapsto\,
         \mathcal{S}_{x'} \cdot \mathscr{V}_{x'}
       \big)
       \;\;\;
       \in
       \;
       \mathbf{C}^{\mathbf{X}}
       \\
       \mathcal{S}_{\mathbf{X}}
       \,\in\, \mathbf{V}^{\mathbf{X}}
       ;\;
       \mathscr{V}_{\mathbf{X}}
       \in\,
       \mathbf{C}^{\mathbf{X}}
       &\vdash&
       \big(\mathscr{V}_{\mathbf{X}}\big)^{
         \mathcal{S}_{\mathbf{X}}
       }
       &
       \coloneqq\;
       \Big(
         x
         \,\mapsto\,
         \textstyle{\int}_{x' \in \mathbf{X}}
         \big(\mathscr{W}_{x'}\big)^{
           \mathbf{X}(x,x')
           \otimes
           \mathcal{S}_{x'}
         }
       \Big)
       \;\;\;
       \in
       \;
       \mathbf{C}^{\mathbf{X}}
       \mathrlap{\,.}
     \end{array}
   $$

\end{proposition}
\begin{proof}
Observe the following sequence of [[natural isomorphisms]]:
$$
  \begin{array}{ll}
  Func(\mathbf{X},\,\mathbf{V})
  \big(
    \mathcal{R}_{\mathbf{X}}
    \;,\;
    [\mathcal{S}_{\mathbf{X}},\, \mathcal{T}_{\mathbf{X}}]
  \big)
  \\
  \;\simeq\;
  \textstyle{\int}_{x \in \mathbf{X}}
  \mathbf{V}
  \Big(
    \mathcal{R}_{x}
    \;,\;
    \int_{y \in \mathbf{X}}
    \mathbf{V}
    \big(
      \mathbf{X}(x,y)
      \otimes
      \mathcal{S}_{y}
      \;,\;
      \mathcal{T}_{y}
    \big)
  \Big)
  &
  \text{by the definitions}
  \\
  \;\simeq\;
  \int_{x \in \mathbf{X}}
  \int_{y \in \mathbf{X}}
  \mathbf{V}
  \Big(
    \mathcal{R}_{x}
    \;,\;
    \mathbf{V}
    \big(
      \mathbf{X}(x,y)
      \otimes
      \mathcal{S}_{y}
      \;,\;
      \mathcal{T}_{y}
    \big)
  \Big)
  &
  \text{ since internal homs preserve limits }
  \\
  \;\simeq\;
  \int_{x \in \mathbf{X}}
  \int_{y \in \mathbf{X}}
  \mathbf{V}
  \Big(
    \mathbf{X}(x,y)
    \otimes
    \mathcal{R}_{x}
    \otimes
    \mathcal{S}_{y}
    \;,\;
    \mathcal{T}_{y}
  \Big)
  &
  \text{by closed monoidal structure of}\;\mathbf{V}
  \\
  \;\simeq\;
  \int_{y \in \mathbf{X}}
  \int_{x \in \mathbf{X}}
  \mathbf{V}
  \Big(
    \mathbf{X}(x,y)
    \otimes
    \mathcal{R}_{x}
    \otimes
    \mathcal{S}_{y}
    \;,\;
    \mathcal{T}_{y}
  \Big)
  &
  \text{ by Fubini theorem for ends }
  \\
  \;\simeq\;
  \int_{y \in \mathbf{X}}
  \mathbf{V}
  \Big(
    \big(
    \int^{x \in \mathbf{X}}
    \mathbf{X}(x,y)
    \otimes
    \mathcal{R}_{x}
    \big)
    \otimes
    \mathcal{S}_{y}
    \;,\;
    \mathcal{T}_{y}
  \Big)
  &
  \text{ since internal homs preserve limits }
  \\
  \;\simeq\;
  \int_{y \in \mathbf{X}}
  \mathbf{V}
  \Big(
    \mathcal{R}_{y}
    \otimes
    \mathcal{S}_{y}
    \;,\;
    \mathcal{T}_{y}
  \Big)
  &
  \text{ by co-Yoneda lemma }
  \\
  \;\simeq\;
  \int_{y \in \mathbf{X}}
  \mathbf{V}
  \Big(
    \big(
    \mathcal{R}_{\mathbf{X}}
    \otimes
    \mathcal{S}_{\mathbf{X}}
    \big)_{y}
    \;,\;
    \mathcal{T}_{y}
  \Big)
  &
  \text{ by definition }
  \\
  \;\simeq\;
  Func(\mathbf{X},\mathbf{V})
  \big(
    \mathcal{R}_{\mathbf{X}}
    \otimes
    \mathcal{S}_{\mathbf{X}}
    \;,\;
    \mathcal{T}_{\mathbf{X}}
  \big)
  &
  \text{ by definition, }
  \end{array}
$$
where we used, apart from the above definitions:

1. that [[internal hom-functors preserve limits]]

1. the [Fubini theorem for ends](end#Fubini)

1. the [[co-Yoneda lemma]].

This establishes that $[-,-]$ is an internal hom in $Func(\mathbf{X},\mathbf{V})$ for the objectwise tensor product, as claimed.

The natural isomorphisms needed to exhibit the (co)tensoring of $\mathbf{C}^{\mathbf{X}}$ both follow essentially the same sequence of steps, just up to the relevant substitutions. For definiteness we spell it out again, but now proceeding in the reverse order of steps:

Given

* $\mathcal{S}_{\mathbf{X}} \in Func(\mathbf{X},\mathbf{V})$,

* $\mathscr{V}_{\mathbf{X}},\, \mathscr{W}_{\mathbf{X}}  \in Func(\mathbf{X},\mathbf{V})$,

* $x \in \mathbf{X}$,

consider the following sequence of [[enriched natural transformation|$V$-enriched natural]] isomorphisms in these variables:

Starting with
$$
  \begin{array}{ll}
    \mathbf{C}^{\mathbf{X}}
    \big(
      \mathcal{S}_{\mathbf{X}}
      \cdot
      \mathscr{V}_{\mathbf{X}}
      \;,\;
      \mathscr{W}_{\mathbf{X}}
    \big)_x
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \mathbf{C}\Big(
      \mathbf{X}(x,z)
      \cdot
      \big(
        \mathcal{S}_{\mathbf{X}}
        \cdot
        \mathscr{V}_{\mathbf{X}}
      \big)_{z}
      ,\,
      \mathscr{W}_{z}
    \Big)
    &
    \text{ by definition }
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \mathbf{C}\big(
      \mathbf{X}(x,z)
      \cdot
      \mathcal{S}_{z}
      \cdot
      \mathscr{V}_{z}
      \;,\;
      \mathscr{W}_{z}
    \big)
    &
    \text{ by definition }
    \\
    \;\simeq\;
    \cdots
  \end{array}
$$
we invoke the [[co-Yoneda lemma]], either to introduce a fresh variable on $\mathbf{X}(x,-) \cdot \mathcal{S}_{(-)}$
$$
  \begin{array}{ll}
    \cdots
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \mathbf{C}\Big(
      \int^{y \in \mathbf{X}}
      \mathbf{X}(y,z)
      \cdot
      \big(
        \mathbf{X}(x,y)
        \cdot
        \mathcal{S}_{y}
      \big)
      \cdot
      \mathscr{V}_{z}
      \;,\;
      \mathscr{W}_{z}
    \Big)
    &
    \text{ co-Yoneda lemma }
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \int_{y \in \mathbf{X}}
    \mathbf{C}\Big(
      \mathbf{X}(y,z)
      \cdot
      \mathbf{X}(x,y)
      \cdot
      \mathcal{S}_{y}
      \cdot
      \mathscr{V}_{z}
      \;,\;
      \mathscr{W}_{z}
    \Big)
    &
    \text{ enriched homs preserve enriched limits }
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \int_{y \in \mathbf{X}}
    \mathbf{V}
    \Big(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{S}_{y}
      \;,\;
      \mathbf{C}\big(
        \mathbf{X}(y,z)
        \cdot
        \mathscr{V}_{z}
        \;,\;
        \mathscr{W}_{z}
      \big)
    \Big)
    &
    \text{ tensoring iso of }\; \mathbf{C}
    \\
    \;\simeq\;
    \int_{y \in \mathbf{X}}
    \int_{z \in \mathbf{X}}
    \mathbf{V}
    \Big(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{S}_{y}
      \;,\;
      \mathbf{C}\big(
        \mathbf{X}(y,z)
        \cdot
        \mathscr{V}_{z}
        \;,\;
        \mathscr{W}_{z}
      \big)
    \Big)
    &
    \text{ Fubini theorem for ends }
    \\
    \;\simeq\;
    \int_{y \in \mathbf{X}}
    \mathbf{V}
    \Big(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{S}_{y}
      \;,\;
      \int_{z \in \mathbf{X}}
      \mathbf{C}\big(
        \mathbf{X}(y,z)
        \cdot
        \mathscr{V}_{z}
        \;,\;
        \mathscr{W}_{z}
      \big)
    \Big)
    &
    \text{ enriched homs preserve enriched limits }
    \\
    \;\simeq\;
    \int_{y \in \mathbf{X}}
    \mathbf{V}\Big(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{S}_{y}
      \;,\;
      \mathbf{C}^{\mathbf{X}}
      \big(
        \mathscr{V}_{\mathbf{X}}
        ,\,
        \mathscr{W}_{\mathbf{X}}
      \big)_y
    \Big)
    &
    \text{ definition }
    \\
    \;\simeq\;
    \mathbf{V}^{\mathbf{X}}\Big(
      \mathcal{S}_{\mathbf{X}}
      \;,\;
      \mathbf{C}^{\mathbf{X}}
      \big(
        \mathscr{V}_{\mathbf{X}}
        ,\,
        \mathscr{W}_{\mathbf{X}}
      \big)
    \Big)_x
    &
    \text{ definition }
  \end{array}
$$
or to get a fresh variable on $\mathbf{X}(x,-) \cdot \mathscr{V}_{(-)}$:
$$
  \begin{array}{ll}
    \cdots
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \mathbf{C}
    \Big(
      \mathcal{S}_z
      \cdot
      \int^{y \in \mathbf{X}}
      \mathbf{X}(y,z)
      \cdot
      \big(
        \mathbf{X}(x,y)
        \cdot
        \mathcal{V}_{y}
      \big)
      \;,\;
      \mathscr{W}_{z}
    \Big)
    &
    \text{ co-Yoneda lemma }
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \int_{y \in \mathbf{X}}
    \mathbf{C}
    \Big(
      \mathbf{X}(x,y)
      \cdot
      \mathbf{X}(y,z)
      \cdot
      \mathcal{S}_z
      \cdot
      \mathcal{V}_{y}
      \;,\;
      \mathscr{W}_{z}
    \Big)
    &
    \text{ enriched homs preserve enriched limits }
    \\
    \;\simeq\;
    \int_{z \in \mathbf{X}}
    \int_{y \in \mathbf{X}}
    \mathbf{C}
    \Big(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{V}_{y}
      \;,\;
      \big(\mathscr{W}_{z}\big)^{
        \mathbf{X}(y,z)
        \cdot
        \mathcal{S}_z
      }
    \Big)
    &
    \text{ cotensoring iso of }\; \mathbf{C}
    \\
    \;\simeq\;
    \int_{y \in \mathbf{X}}
    \int_{z \in \mathbf{X}}
    \mathbf{C}
    \Big(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{V}_{y}
      \;,\;
      \big(\mathscr{W}_{z}\big)^{
        \mathbf{X}(y,z)
        \cdot
        \mathcal{S}_z
      }
    \Big)
    &
    \text{ Fubini theorem for ends }
    \\
    \;\simeq\;
    \int_{y \in \mathbf{X}}
    \mathbf{C}
    \Big(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{V}_{y}
      \;,\;
      \int_{z \in \mathbf{X}}
      \big(\mathscr{W}_{z}\big)^{
        \mathbf{X}(y,z)
        \cdot
        \mathcal{S}_z
      }
    \Big)
    &
    \text{ enriched homs preserve enriched limits }
    \\
    \;\simeq\;
    \int_{y \in \mathbf{X}}
    \mathbf{C}
    \bigg(
      \mathbf{X}(x,y)
      \cdot
      \mathcal{V}_{y}
      \;,\;
      \Big(
        \big(
          \mathscr{W}_{\mathbf{X}}
        \big)^{
          \mathcal{S}_{\mathbf{X}}
        }
      \Big)_y
    \bigg)
    &
    \text{ definition }
    \\
    \;\simeq\;
    \mathbf{C}^{\mathbf{X}}
    \Big(
      \mathcal{V}_{\mathbf{X}}
      \;,\;
      \big(\mathscr{W}_{\mathbf{X}}\big)^{
        \mathcal{S}_{\mathbf{X}}
      }
    \Big)_x
    &
    \text{ definition }
  \end{array}
$$

By naturality in $x \in \mathbf{X}$, these constitute the required $\mathbf{V}^{\mathbf{X}}$-[[enriched natural transformation|-enriched]] [[natural isomorphisms]] for exhibiting the claimed (co)tensoring:
$$
  \mathbf{C}^{\mathbf{X}}
  \big(
    \mathcal{S}_{\mathbf{X}}
    \cdot
    \mathscr{V}_{\mathbf{X}}
    \;,\;
    \mathscr{W}_{\mathbf{X}}
  \big)
  \;\simeq\;
  \mathbf{V}^{\mathbf{X}}
  \Big(
    \mathcal{S}_{\mathbf{X}}
    \;,\;
    \mathbf{C}^{\mathbf{X}}\big(
      \mathscr{V}_{\mathbf{X}}
      \;,\;
      \mathscr{W}_{\mathbf{X}}
    \big)
  \Big)
  \;\simeq\;
  \mathbf{C}^{\mathbf{X}}
  \Big(
    \mathscr{V}_{\mathbf{X}}
    \;,\;
    \big(\mathscr{W}_{\mathbf{X}}\big)^{
      \mathcal{S}_{\mathbf{X}}
    }
  \Big)
  \,.
$$
From this, finally, follows an $\mathbf{V}^{\mathbf{X}}$-enriched composition operation
$$
  \mathbf{C}^{\mathbf{X}}\big(
    \mathscr{R}_{\mathbf{X}}
    ,\,
    \mathscr{V}_{\mathbf{X}}
  \big)
  \otimes
  \mathbf{C}^{\mathbf{X}}\big(
    \mathscr{V}_{\mathbf{X}}
    ,\,
    \mathscr{W}_{\mathbf{X}}
  \big)
  \longrightarrow
  \mathbf{C}^{\mathbf{X}}\big(
    \mathscr{R}_{\mathbf{X}}
    ,\,
    \mathscr{W}_{\mathbf{X}}
  \big)
$$
to be defined as the [[adjunct]] (via the just established adjunctions) of the consecutive [[evaluation maps]] (being themselves the respective [[counit of an adjunction|adjunction counits]]):
$$
  \mathscr{R}_{\mathbf{X}}
  \cdot
  \mathbf{C}^{\mathbf{X}}\big(
    \mathscr{R}_{\mathbf{X}}
    ,\,
    \mathscr{V}_{\mathbf{X}}
  \big)
  \cdot
  \mathbf{C}^{\mathbf{X}}\big(
    \mathscr{V}_{\mathbf{X}}
    ,\,
    \mathscr{W}_{\mathbf{X}}
  \big)
  \overset{ev}{\longrightarrow}
  \mathscr{V}_{\mathbf{X}}
  \cdot
  \mathbf{C}^{\mathbf{X}}\big(
    \mathscr{V}_{\mathbf{X}}
    ,\,
    \mathscr{W}_{\mathbf{X}}
  \big)
  \overset{ev}{\longrightarrow}
  \mathscr{W}_{\mathbf{X}}
  \,.
$$
\end{proof}

## Related concepts

* [[enriched Reedy category]]

* [[model structure on homotopical presheaves]]

* [[model structure on sSet-presheaves]]

## References

An original article:

* [[Brian Day]], [[Max Kelly]], *Enriched functor categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106** (1969) 178-191 [[doi:10.1007/BFb0059146](https://doi.org/10.1007/BFb0059146), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0059146.pdf)]

Review includes

* {#Day70} [[Brian Day]], _On closed categories of functors_, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/DayReport.pdf))

* {#Kelly82} [[Max Kelly]], [section 2.2 p. 29](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf#page=29) of: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

[[!redirects enriched functor categories]]

[[!redirects enriched-functor category]]
[[!redirects enriched-functor categories]]

[[!redirects enriched functor-category]]
[[!redirects enriched functor-categories]]

[[!redirects enriched presheaf category]]
[[!redirects enriched presheaf categories]]

[[!redirects category of enriched presheaves]]
[[!redirects categories of enriched presheaves]]


enriched groupoid

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of *enriched groupoids* is the generalization to [[enriched category theory]] of the notion of *[[groupoids]]* from plain [[category theory]].

## Definition

Let $(\mathcal{V}, \times, \ast)$ be a *[[cartesian monoidal category]]* (serving as an [[base of enrichment]]), so that, in particular,

* the [[tensor unit]] is a [[terminal object]] $\ast$,

* for each [[object]] $v \in \mathcal{V}$ there is a [[diagonal morphism]]

  $$
    \Delta_v \,\colon\, v \to v \times v
    \,.
  $$

A *$\mathcal{V}$-enriched groupoid is a $\mathcal{V}$-[[enriched category]] $\mathcal{C}$ equipped for $X, Y \,\in\, Obj(\mathcal{C})$ with [[morphisms]] (in $\mathcal{V}$) of the form

$$
  inv_{X,Y}
  \;\colon\;
  \mathcal{C}(X,Y)
  \longrightarrow
  \mathcal{C}(Y,X)
$$

which behave like assigning [[inverse morphisms]] in that the [[composition]] morphisms of $\mathcal{C}$

$$
  comp_{X, Y, Z}
  \;\colon\;
  \mathcal{C}(X,Y)
  \otimes
  \mathcal{C}(Y,Z)
  \longrightarrow
  \mathcal{C}(Y,Z)
$$

satisfy

$$
  comp_{Y,X,Y}
    \circ
  \big(
    inv_{X,Y}
      \times
    id_{\mathcal{C}(X,Y)}
  \big)
    \circ
  \Delta_{\mathcal{C}(X,Y)}
  \;\;=\;\;
  \mathcal{C}(X,Y)
  \overset{}{\longrightarrow}
  \ast
  \overset{id_Y}{\longrightarrow}
  \mathcal{C}(Y,Y)
$$

and

$$
  comp_{X,Y,X}
    \circ
  \big(
    id_{\mathcal{C}(X,Y)}
      \times
    inv_{X,Y}
  \big)
  \circ
  \Delta_{\mathcal{C}(X,Y)}
  \;\;=\;\;
  \mathcal{C}(X,Y)
  \overset{}{\longrightarrow}
  \ast
  \overset{id_X}{\longrightarrow}
  \mathcal{C}(X,X)
  \,.
$$

## Examples

\begin{example}
  For [[Set]] regarded with its [[cartesian monoidal category|cartesian monoidal]] [[structure]], [[Set]]-enriched groupoids are ordinary [[groupoids]].
\end{example}

\begin{example}
\label{StrictTwoGroupoids}
For [[Grpd]] regarded as a [[1-category]] and equipped with its [[cartesian monoidal category|cartesian monoidal]] [[structure]] given by forming [[product categories|product groupoids]], [[Grpd]]-enriched groupoids are [[strict 2-groupoids]].
\end{example}

\begin{example}\label{SimplicialGroupoid}
  The [[sSet]]-enriched groupoids are traditionally  misnamed *[[simplicial groupoids]]* (following [Dwyer & Kan (1984)](simplicial+groupoid#DwyerKan84) and similar abuse for *[[simplicial category]]*), see there for more. More pedantically, $sSet$-enriched groupoids are only those [[sSet]]-[[internal groupoids]] whose set of objects is constant, cf. Exp. \ref{RelationToInternalGroupoids} below.

Moreover, [[fundamental infinity-groupoid|fundamental $\infty$-groupoids]] incarnated as [[sSet]]-[[enriched groupoids]] aka "Dwyer-Kan [[simplicial groupoids]]" are (mis-)named *[[Dwyer-Kan loop groupoids]]*.
\end{example}

\begin{example}
  Generally, a $\mathcal{V}$-enriched groupoid with a single object is equivalently the [[delooping groupoid]] of a [[group object]] [[internalization|internal]] to $\mathcal{V}$.

For instance, a one-object [[sSet]]-groupoid (Exp. \ref{SimplicialGroupoid}) is the [[delooping groupoid]] of a [[simplicial group]].
\end{example}

\begin{example}\label{RelationToInternalGroupoids}
  An [[internal groupoid]] in $\mathcal{V}$ is equivalently a $\mathcal{V}$-enriched groupoid if its $\mathcal{V}$-object of objects is in the image of the [[coproduct]]-[[preserved colimit|preserving]] [[functor]] $Set \to \mathcal{V}$ (using here the assumption that $\mathcal{V}$ is a [[Bénabou cosmos]], hence [[cocomplete category|cocomplete]], hence with all [[coproducts]]).
\end{example}

## Related concepts

* [[enriched category]]

* [[internal groupoid]], [[internal group]]

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

## References
 {#References}

The notion of enriched groupoids is [[folklore]], originating in the special case of enrichment over [[sSet]], where it is traditionally discussed, going back to [Dwyer & Kan (1980)](simplicial+groupoid#DwyerKan80), [(1984)](simplicial+groupoid#DwyerKan84), in the guise of [[simplicial objects]] in [[Grpd]] with discrete simplicial set of objects and then referred to (inaccurately) as *[[simplicial groupoids]]* (see there for more).

Later

* [[John F. Jardine]], §9.3 in: *[[Local homotopy theory]]*, Springer Monographs in Mathematics (2015) [[doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)]

switches to the terminology of *enriched groupoids* over simplicial sets (but still does not give the definition of enriched groupoids).

References that make the general definition of enriched groupoids explicit:

* [[Jacopo Emmenegger]], [[Fabio Pasquali]], [[Giuseppe Rosolini]], §3 in: *Elementary fibrations of enriched groupoids*, Mathematical Structures in Computer Science **31** 9 (2021) 958-978 [[doi:10.1017/S096012952100030X](https://doi.org/10.1017/S096012952100030X)]

  > (discussed as [[categorical semantics]] for [[homotopy type theory]])

[[!redirects enriched groupoids]]


enriched hom-functor

[[!redirects enriched hom-functors]]
[[!redirects enriched hom-functors]]

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## Idea

The concept of _enriched hom-functor_ is the generalization of that of [[hom-functors]] from [[category theory]] to [[enriched category theory]].

## Definition

+-- {: .num_defn}
###### Definition
**(enriched hom-functor)**

For $\mathcal{V}$ a [[closed monoidal category|closed]] [[symmetric monoidal category]] with all [[limits]] and [[colimits]], let $\mathcal{C}$, be a $\mathcal{V}$-[[enriched category]]. Then its _$\mathcal{V}$-enriched hom-functor_ is the $\mathcal{V}$-[[enriched functor]]

$$
  \mathcal{C}(-,-)
  \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
  \longrightarrow
  \mathcal{V}
$$

from the [[enriched product category]] of $\mathcal{C}$ with its enriched [[opposite category]] to $\mathcal{V}$, canonically regarded as [[enriched category|enriched]] over itself, which sends [[pairs]] of [[objects]] to the corresponding [[hom-object]] in $\mathcal{C}$

$$
  c_1,c_2 \mapsto \mathcal{C}(c_1,c_2)
$$

and which on [[hom-objects]] of $\mathcal{C}^{op} \times \mathcal{C}$ is given by the evident pre- and post-[[composition]] operation.

=--

(e.g. [Borceux 94, Prop. 6.2.7](#Borceux94))

## Related concepts

* [[hom-set]], [[hom-object]]

* [[hom-functor]]

* [[enriched hom-functor]], [[internal hom-functor]]

* [[derived hom-functor]]

## References

* {#Borceux94} [[Francis Borceux]], Vol 2, Prop. 6.2.7 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)


enriched homotopical category

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#Contents#
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## Idea ##

The concept of _enriched homotopical category_ is the generalization of the concept of [[homotopical category]] to the context of [[enriched category theory]] and hence to [[homotopy coherent category theory]].

The idea is that the [[homotopy category]] $Ho_C$ of a category $C$ which is enriched over a suitable [[monoidal category|monoidal]]  and [[homotopical category|homotopical]] category $V$ is itself a $Ho_V$-enriched category.

## Definition ##

For $V$ a [[closed monoidal homotopical category]],
a $V$-[[enriched category]] $C$ with [[power|powers]] and [[copower|copowers]] and with the structure of a [[homotopical category]] on its underlying category $C_0$ is a **$V$-homotopical category** when equipped with a [[deformation retract for the enrichment]].

## Examples ##

If $V$ is a [[monoidal model category]], then any $V$-[[enriched model category]] is automatically a $V$-homotopical category.

## Consequences ##

[[closed monoidal homotopical category|Recall]] that for $V$ as above, $Ho_V$ is closed monoidal.

+-- {: .un_prop}
###### Proposition
With $C$ a $V$-homotopical category, $Ho_{C_0}$ is the underlying category of a $Ho_V$-enriched category.
=--

Write $Ho_C$ for this $Ho_V$-enriched category.  This is the enriched analogue of the [[homotopy category]] of $C$.

So schematically we have (with all of the above qualifiers suppressed):
$$
  (C \in V-Cat) \Rightarrow (Ho_C \in Ho_V-Cat)
$$

## Construction of the enriched homotopy category ##

For $C$ an enriched homotopical $V$-category as above, the  $Ho_V$-category $Ho_C$ is constructed from the homotopy category $Ho_{C_0}$ of the ordinary category underlying $C$ by constructing a $Ho_V$-[[closed monoidal category module|module structure]], essentially following
section 4.3.2 of Hovey: _Model categories_.

## References ##

The definition appears as [definition 16.1, p. 46](http://arxiv.org/PS_cache/math/pdf/0610/0610194v1.pdf#page=46) in Shulman: [Homotopy limits and colimits in enriched homotopy theory](http://arxiv.org/abs/math.AT/0610194v1), the proposition is [proposition 16.2, p. 46](http://arxiv.org/PS_cache/math/pdf/0610/0610194v1.pdf#page=46). The construction of $Ho_C$ follows the proof of [proposition 15.4, p. 45](http://arxiv.org/PS_cache/math/pdf/0610/0610194v1.pdf#page=46).


enriched homotopy theory

#Idea#

_Enriched homotopy theory_ is about combining the notion of categories that model aspects of [[homotopy theory]] (such as [[homotopical category|homotopical categories]] and [[model category|model categories]]) with the ideas and tools of [[enriched category theory]].

On the one hand, enriched category theory is already used in many places in ordinary homotopy theory, usually for the particular case of enrichment over [[topological space]]s or [[simplicial set]]s.  Just as an ordinary category has a _set_ of morphisms between any two objects, an ordinary homotopy theory has a _space_ of morphisms between any two objects (including morphisms, homotopies between them, higher homotopies, and so on).  From a [[higher category theory|higher categorical]] viewpoint, a homotopy theory is a presentation of an $(\infty,1)$-[[(infinity,1)-category|category]], and $(\infty,1)$-categories can be viewed as categories enriched (perhaps weakly) over $\infty$-groupoids (that is, spaces).  Many of the model categories used in homotopy theory come with natural topological or simplicial enrichments that make them better-behaved and easier to work with.  The enrichment can also be used in a detailed study of homotopy coherence, without necessarily needing model-categorical tools.  See [[homotopy coherent category theory]].

On the other hand, instead of using enriched category theory as a tool to understand ordinary homotopy theory, one can look for a common generalization of them both.  Now we take the point of view that spaces (or $\infty$-groupoids, or simplicial sets) are the higher-categorical counterpart of _sets_, and look for an analogue of the passage from Set-enriched categories to $V$-enriched categories that starts from ordinary (space-enriched) homotopy theories and ends up at $V$-enriched homotopy theories.  Note that now $V$ need not be any space-like category, so that $V$-enriched homotopy theories need not be very similar to $(\infty,1)$-categories.  One use of this approach is to get at a sort of higher categories with noninvertible $k$-cells for $k\gt 1$, by taking $V$ to be [[folk model structure|categories]], 2-categories, and so on (perhaps [[Trimble n-category|iteratively]]).  But $V$ could also be [[chain complex]]es, or $W$-categories for some other $W$, or [[spectrum|spectra]], or some other category with even less resemblance to "spaces."

Fortunately, both of these approaches can be handled by many of the same technical tools, including but not limited to:

* [[monoidal model category|Monoidal model categories]] and [[enriched model category|enriched model categories]].

* [[closed monoidal homotopical category|Closed monoidal homotopical categories]] and [[enriched homotopical category|enriched homotopical categories]].

* [[operad|Operads]] and $A_\infty$-[[A-infinity-category|categories]].


enriched Lawvere theory

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# Contents
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## Idea

The notion of **enriched Lawvere theory** or **enriched algebraic theory** is a generalization of [[Lawvere theories]] to the setting of [[enriched categories]].

## Definitions and elementary properties

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

An **enriched [[Lawvere theory]]** over a [[locally finitely presentable]] $V$-[[enriched category]] $A$ is a small $V$-[[enriched category]] $L$ together with a $V$-[[enriched functor]] $J\colon A_f^{op}\to L$
that induces an identity map on the set of objects.
Here $A_f$ denotes the [[full subcategory]] of [[finitely presentable objects]] in $A$.
The category $V$ is a locally finitely presentable [[closed symmetric monoidal category]].

=--

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

An **algebra** over an enriched Lawvere theory is
an object $X$ in $A$ equipped with a functor $M\colon L\to V$ whose precomposition with $J$ is isomorphic to the representable functor of $X$.

=--

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

For the case $A=Set$ the above definition recovers
the usual notion of a [[Lawvere theory]]
and an [[algebra]] over a Lawvere theory.

=--

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

The forgetful functor from algebras to $A$
is a [[finitary enriched monadic functor]] (over $V$).

=--

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

The category of enriched Lawvere theories over $A$
is equivalent to the category of [[finitary enriched
monads]] over $A$.
The corresponding $V$-enriched categories of
algebras are also equivalent.

=--

## References

* [[Eduardo Dubuc]], _Enriched semantics-structure (meta) adjointness_, Rev. Un. Mat. Argentina 25 (1970): 5-26.

* [[John Power]], _Enriched Lawvere theories_, [tac](http://www.tac.mta.ca/tac/volumes/6/n7/6-07abs.html)

* [[Koki Nishizawa]], [[John Power]], _Lawvere theories enriched over a general base_. Journal of Pure and Applied Algebra **213**, Issue 3, March 2009, Pages 377--386. ([pdf](https://www.sciencedirect.com/science/article/pii/S0022404908001485)),
MR2477057, Zbl:1158.18003,
doi:[10.1016/j.jpaa.2008.07.009](http://dx.doi.org/10.1016/j.jpaa.2008.07.009).

* [[Sam Staton]], _Freyd categories are enriched Lawvere theories_, [pdf](http://www.cs.ox.ac.uk/people/samuel.staton/papers/freyd-lawvere-2014.pdf)

* Rory B. B. Lucyshyn-Wright, _Enriched algebraic theories and monads for a system of arities_, ([arXiv:1511.02920](https://arxiv.org/abs/1511.02920)).

* [[Stephen Lack]], [[John Power]], _Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base_, [pdf](http://maths.mq.edu.au/~slack/papers/jfp.pdf)

* [[Richard Garner]], [[John Power]], _An enriched view on the extended finitary monad--Lawvere theory correspondence_, ([ arXiv:1707.08694](https://arxiv.org/abs/1707.08694))

* {#Arkor22Thesis} [[Nathanael Arkor]], *Monadic and Higher-Order Structure*, PhD thesis, Cambridge (2022) [[doi:10.17863/CAM.86347](https://doi.org/10.17863/CAM.86347), [pdf](https://www.repository.cam.ac.uk/bitstreams/666be4fb-957b-4c2e-83fc-8124621f0c43/download), [[Arkor-MonadicAndHigherStructure.pdf:file]]]

[[!redirects enriched Lawvere theories]]
[[!redirects enriched algebraic theory]]
[[!redirects enriched algebraic theories]]


enriched model category

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#Contents#
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{:toc}

## Idea

An _enriched model category_ is an [[enriched category]] $C$ together with the structure of a [[model category]] on the [underlying category](http://ncatlab.org/nlab/show/enriched+category#BaseChange) $C_0$ such that both structures are compatible in a reasonable way.

## Definition

Let $V$ be a [[monoidal model category]].

A **$V$-enriched model category** is

* a $V$-[[enriched category]] $C$

  * which is [[powered and copowered category|powered and copowered]] over $V$;

* with the structure of a [[model category]] on the [underlying category](enriched+category#BaseChange) $C_0$

* such that

  * {#PullbackPowerAxiom} **([[pullback-power axiom]])** for every [[cofibration]] $i \colon A \to B$ and [[fibration]] $p \colon X \to Y$ in $C_0$ the [[pullback powering]] morphism (dual to the [[pushout product]]) with respect to the powering in $V$

    $$
      C(B,X) \stackrel{(i^* , p_*)}{\to} C(A,X) \times_{C(A,Y)} C(B,Y)
    $$

    is a fibration with respect to the model structure on $V$;

  * and is an [[acyclic fibration]] whenever $i$ or $p$ are acyclic.

The last two conditions here are equivalent to the fact that the [[copower]]

$$
  \otimes : C \times V \to C
$$

is a [[Quillen bifunctor]].

## Properties

### Change of enrichment

(...)

## Examples

\begin{example}\label{MonoidalModelCategoryIsEnrichedModelOverItself}
**([[monoidal model category]] is enriched model over itself)**
\linebreak
  Every [[monoidal model category]] is an [[enriched model category]] over itself, via the [[enriched category|enrichment]] of its underlying [[closed monoidal category]].
\end{example}
\begin{proof}
  One just needs to see that the [[pullback-power axiom]] is implied by (in fact it is equivalent to) the [[pushout-product axiom]]. This equivalence is an instance of [[Joyal-Tierney calculus]] (see [this Prop.](Introduction+to+Homotopy+Theory#JoyalTierneyCalculus)):

Writing

* $\mathrm{F}$, $\mathrm{C}$, $\mathrm{W}$ for the classes of [[fibrations]], [[cofibrations]], [[weak equivalences]], respectively;

* $\mathrm{FW}$, $\mathrm{CW}$ for the classes of [[acyclic fibrations]] and [[acyclic cofibrations]], respectively

and

* $(-) \,⧄\, (-)$ for the [[lifting property]],

* $(-) \Box (-)$ for the [[pushout product]],

* $(-)^{\Box (-)}$ for the [[pullback power]],

we have the following logical equivalences.

$$
  \begin{aligned}
    \mathrm{C} \Box \mathrm{C}
    &
    \,\subset\,
    &
    \mathrm{C}
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{C} \Box \mathrm{C}
    &\,\;⧄\;\,&
    \mathrm{FW}
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{C}
    &
    \,⧄\,
    &
    \mathrm{FW}^{\Box \mathrm{C}}
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{FW}^{\Box \mathrm{C}}
    &
    \,\subset\,
    &
    \mathrm{FW}
    \\
    {\phantom{-}}
    \\
    \mathrm{C} \Box \mathrm{CW}
    &
    \,\subset\,
    &
    \mathrm{CW}
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{C} \Box \mathrm{CW}
    &\,\;⧄\;\,&
    \!\!\mathrm{F}\;\;\;
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{C}
    &
    \;⧄\;
    &
    \mathrm{F}^{\Box \mathrm{CW}}
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{F}^{\Box \mathrm{CW}}
    &
    \,\subset\,
    &
    \mathrm{FW}
    \\
    &
    &
    &
    &
    &&
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{CW}
    &
    \;⧄\;
    &
    \mathrm{F}^{\Box \mathrm{C}}
    &
    \;\;\;
    \Leftrightarrow
    \;\;\;
    &
    \mathrm{F}^{\Box \mathrm{C}}
    &
    \,\subset\,
    &
    \mathrm{F}
  \end{aligned}
$$

Here the outer equivalences are by definition of the [[lifting properties]] in a [[model category]], while the middle equivalences are by [[Joyal-Tierney calculus]].
The statements on the far left constitute the [[pushout-product axiom]], while those on the far right constiture the [[pullback-power axiom]].
\end{proof}

\begin{example}\label{TheClassicalModelCategoriesOfTopSpacesAndSimpSets}
  Since the [[model structure on compactly-generated topological spaces]] as well as the [[classical model structure on simplicial sets]] are [[monoidal model categories]], they are,
by by Exp. \ref{MonoidalModelCategoryIsEnrichedModelOverItself},
also enriched modal categories over themselves.
\end{example}

\begin{example}
  A model category enriched over the [[classical model structure on simplicial sets]] is called a *[[simplicial model category]]*.
\end{example}

## Related concepts

* [[monoidal model category]]

* [[enriched monoidal model category]]

* [[derived hom-functor]]

* [[enriched homotopical category]].

* [[pushout-product]], [[pushout-product  axiom]]

* [[pullback power]]

## References

The general definition of enriched model categories has its origin in

* {#Quillen67} [[Daniel Quillen]], chapter II, section 2 of: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics __43__, Springer (1967) [[doi:10.1007/BFb0097438](https://link.springer.com/book/10.1007/BFb0097438)]

where it is formulated in the special case of [[simplicial model categories]] (but the generalization is immediate).

Textbook account:

* [[Jacob Lurie]], Def. A.3.1.5 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

See also:

* [[Bertrand Guillou]], [[Peter May]], _Enriched model categories and presheaf categories_, New York J. Math. 26 (2020) 37–9 ([arXiv:1110.3567](http://arxiv.org/abs/1110.3567), [NYJM:2020/26-3](https://www.emis.de/journals/NYJM/j/2020/26-3.html))

On enriched [[Reedy model categories]]:

* {#Angeltveit} [[Vigleik Angeltveit]], *Enriched Reedy categories*, Proceedings of the American Mathematical Society **136** 7 (2008)  [[arXiv:math/0612137](http://arxiv.org/abs/math/0612137), [AMS:S0002-9939-08-09185-5](https://www.ams.org/journals/proc/2008-136-07/S0002-9939-08-09185-5/home.html), [jstor:20535419](https://www.jstor.org/stable/20535419)]

[[!redirects enriched model categories]]

[[!redirects enriched model structure]]
[[!redirects enriched model structures]]

[[!redirects enriched model category theory]]