The notion of enriched category is a generalization of the notion of category.
Very often instead of merely having a set of morphisms 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 and generalize it by replacing the hom-sets by hom-objects , which are objects in a suitable category $K$. This gives the concept of ‘enriched category’.
The category $K$ must be monoidal, so that we can define composition as a morphism
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’
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. 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.
Ordinarily enriched categories have been considered as enriched over a monoidal category. This is discussed in the section
More generally, one may think of a monoidal category as a bicategory with a single object and this way regard enrichment in a monoidal category as the special case of enrichment in a bicategory . This is discussed in the section
Enriched categories and enriched functors between them form themselves a category, the category of V-enriched categories.
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 element
such the following diagrams commute:
for all $a,b,c,d \in Obj(C)$:
this says that composition in $C$ is associative;
and
this says that composition is unital.
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
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 spans 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
Under the composition law, we get a natural map
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
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.
Let $B$ be a bicategory, and write $\otimes$ for horizontal (1-cell) composition (written in Leibniz order). A category enriched in the bicategory $B$ consists of a set $X$ together with
such that the associativity and unitality diagrams, as written above, commute. Viewing a monoidal category $M$ as a 1-object bicategory $\Sigma M$, the notion of enrichment in $M$ coincides with the notion of enrichment in the bicategory $\Sigma M$.
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.)
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
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:
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.
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 or 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
These two operations form adjoint functors relating the 2-category Cat to the 2-category $K$-Cat.
More generally, any (lax) monoidal functor $F: K \to L$ between monoidal categories can be regarded as a “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 base” are explored in Geoff Cruttwell’s thesis.
The idea of enriched categories is not unrelated to that of internal categories, but is different. One difference is that in a $K$-enriched category, the objects still form a set (or a proper class) while the arrows are replaced by objects of $K$, while in a category internal to $K$, both the set of objects and the set of arrows are replaced by objects of $K$.
Another difference is that for $K$-enriched categories, $K$ can be any monoidal category, while for $K$-internal categories, it must have pullbacks, which can be thought of as a generalization of cartesian monoidal structure. In particular, a $K$-internal category with one object (that is, whose object-of-objects is a terminal object) is a monoid in $K$ with respect to the cartesian product, whereas a one-object $K$-enriched category is a monoid in $K$ with respect to whatever monoidal structure we use to define enriched categories.
Nevertheless, internalization and enrichment are related in several ways. On the one hand, internal categories and enriched categories are both instances of monads in bicategories (the bicategory of spans and the bicategory of matrices, respectively). On the other hand, when $K$ is an $\infty$-extensive category, such as Set or simplicial sets (or more generally any Grothendieck topos), (small) $K$-enriched categories can be identified with $K$-internal categories whose object-of-objects is discrete (that is, a coproduct of copies of the terminal object).
The page here mentions internalization and enrichment being the result of applying two different interpretation techniques to the same theory.
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 simplicial category, and these form one model for (∞,1)-categories.
Beware: the term ‘simplicial category’ is also used to mean a category 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 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 strict n-category is a category enriched over strict $(n-1)$-categories. In the limit $n \to \infty$ this leads to 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 poset is a category enriched over the category of truth values, where $\otimes$ is conjunction.
An 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.
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 here.
Lawvere theories can be represented as enriched categories as well; see Garner 2013 and Garner and Power 2017 for details.
A standard reference on enriched categories is
Also
Francis Borceux, I. Stubbe, Short introduction to enriched categories (pdf)
Emily Riehl, chapter 3 Basics of enriched category theory in Categorical Homotopy Theory
Vista of some modern generalizations is in
Change of base is discussed in
Geoff Cruttwell, Normed Spaces and the Change of Base for Enriched Categories, pdf
John Armstrong: Enriched categories.
Lawvere theories, and finitary monads, are identified with certain enriched categories and the passage from one to the other with a cocompletion, in
Richard Garner, Lawvere theories, finitary monads and Cauchy-completion, 2013, arxiv
Richard Garner and John Power, An enriched view on the extended finitary monad–Lawvere theory correspondence, 2017, arxiv