nLab
enriched category

Idea

The idea of an enriched category is that we take the definition of locally small category and replace the hom-sets by objects in some monoidal category $K$. So, a category enriched over $K$ (also called a category enriched in $K$, or simply a $K$-category), say $C$, has a collection $\mathrm{ob}(C)$ of objects and for each pair $x,y\in \mathrm{ob}(C)$, a ‘hom-object’

$$\mathrm{hom}(x,y)\in K.$$hom(x,y) \in K .

We then mimic the usual definition of category. In particular, composition is a morphism in $K$:

$$\circ :\mathrm{hom}(y,z)\otimes \mathrm{hom}(x,y)\to \mathrm{hom}(x,z)$$\circ : hom(y,z) \otimes hom(x,y) \to hom(x,z)

where $\otimes$ is the tensor product in $K$.

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 an fc-multicategory.

See also enriched category theory.

Definition

Let $V$ be a monoidal category with

• tensor product $\otimes :C\times C\to C$;

• tensor unit $I\in \mathrm{Obj}(V)$;

• 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 $\mathrm{Obj}(C)$ – called the set of objects;

• for each ordered pair $(a,b)\in \mathrm{Obj}(C)\times \mathrm{Obj}(C)$ of objects in $C$ an object $C(a,b)\in \mathrm{Obj}(V)$ – called the hom-object or object of morphisms from $a$ to $b$;

• for each ordered triple $(a,b,c)$ of objects of $V$ 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 $a\in \mathrm{Obj}(V)$ a morphism $j_a:I\to C(a,a)$ – called the identity element

• such the following diagrams commute:

for all $a,b,c,d\in \mathrm{Obj}(V)$:

$$\begin{array}{ccccc}(C(c,d)\otimes C(b,c))\otimes C(a,b)& & \stackrel{\alpha }{\to }& & C(c,d)\otimes (C(b,c)\otimes C(a,b))\\ {\downarrow }^{{\circ }_{b,c,d}\otimes {\mathrm{Id}}_{C(a,b)}}& & & & {\downarrow }^{{\mathrm{Id}}_{C(c,d)\otimes {\circ }_{a,b,c}}}\\ C(b,d)\otimes C(a,b)& \stackrel{{\circ }_{a,b,c}}{\to }& C(a,d)& \stackrel{{\circ }_{a,c,d}}{\leftarrow }& C(c,d)\otimes C(a,c)\end{array}$$\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,c}}{\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

$$\begin{array}{ccccc}C(b,b)\otimes C(a,b)& \stackrel{{\circ }_{a,b,b}}{\to }& C(a,b)& \stackrel{{\circ }_{a,a,b}}{\leftarrow }& C(a,b)\otimes C(a,a)\\ {\uparrow }^{j_b\otimes {\mathrm{Id}}_{C(a,b)}}& {\nearrow }_l& & {}_r\nwarrow & {\uparrow }^{{\mathrm{Id}}_{C(a,b)}\otimes j_a}\\ I\otimes C(a,b)& & & & C(a,b)\otimes I\end{array}$$\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.

Passage between ordinary categories and enriched categories

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

If $K(I,-):K\to \mathrm{Set}$ has a left adjoint $-\cdot I:\mathrm{Set}\to K$ (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

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

More generally, a (lax) monoidal functor $F:K\to L$ between monoidal categories can be regarded as a “change of base”, so that by applying $F$, any category enriched over $K$ can be seen as enriched over $L$.

Internalization versus Enrichment

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 $\mathrm{\infty }$-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).

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 a simplicial category, and are one model for $(\mathrm{\infty }\mathrm{,1})$-categories.

• Categories enriched in Top are also a model for $(\mathrm{\infty }\mathrm{,1})$-categories.

• 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 $n\to \mathrm{\infty }$ this leads to strict omega-categories.

• A ringoid is a category enriched over Ab.

• An algebroid is a category enriched over Vect.

(In all these cases the standard monoidal structure on the monoidal categories is understood.)

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

• An ultrametric space is a category enriched over the poset $([0,\mathrm{\infty }],\ge )$ 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 group torsor (over a group $G$) can 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.

References

The standard reference on enriched categories is

• G. M. Kelly, Basic concepts of enriched category theory (tac ,pdf)

Blog entries

• John Armstrong: Enriched categories.