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 KK. This gives the concept of ‘enriched category’.

The category KK must be monoidal, so that we can define composition as a morphism

∘:hom(y,z)⊗hom(x,y)→hom(x,z) \circ : hom(y,z) \otimes hom(x,y) \to hom(x,z)

So, a category enriched over KK (also called a category enriched in KK, or simply a KK-category), say CC, has a collection ob(C)ob(C) of objects and for each pair x,y∈ob(C)x,y \in ob(C), a ‘hom-object’

hom(x,y)∈K. hom(x,y) \in K .

We then mimic the usual definition of category.

We may similarly define a functor enriched over KK and a natural transformation enriched over KK, obtaining a strict 2-category of KK-enriched categories, K K -Cat. By general 2-category theory, we thereby obtain notions of KK-enriched adjunction, KK-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 KK-Cat).

More generally, we may allow KK to be a multicategory, a bicategory, a double category, or a virtual double category.

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

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

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

Enrichment in a monoidal category

Let VV be a monoidal category with

tensor product ⊗:V×V→V\otimes : V \times V \to V;
tensor unit I∈Obj(V)I \in Obj(V);
associator α a,b,c:(a⊗b)⊗c→a⊗(b⊗c)\alpha_{a,b,c} : (a \otimes b)\otimes c \to a \otimes (b \otimes c);
left unitor l a:I⊗a→al_a : I \otimes a \to a;
right unitor r a:a⊗I→ar_a : a \otimes I \to a.

A (small) VV-category CC (or VV-enriched category or category enriched over/in VV) is

a set Obj(C)Obj(C) – called the set of objects;
for each ordered pair (a,b)∈Obj(C)×Obj(C)(a,b) \in Obj(C) \times Obj(C) of objects in CC an object C(a,b)∈Obj(V)C(a,b) \in Obj(V) – called the hom-object or object of morphisms from aa to bb;
for each ordered triple (a,b,c)(a,b,c) of objects of CC a morphism ∘ a,b,c:C(b,c)⊗C(a,b)→C(a,c)\circ_{a,b,c} : C(b,c) \otimes C(a,b) \to C(a,c) in VV – called the composition morphism;
for each object a∈Obj(C)a \in Obj(C) a morphism j a:I→C(a,a)j_a : I \to C(a,a) – called the identity element
such the following diagrams commute:

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

(C(c,d)⊗C(b,c))⊗C(a,b) →α C(c,d)⊗(C(b,c)⊗C(a,b)) ↓ ∘ b,c,d⊗Id C(a,b) ↓ Id C(c,d)⊗∘ a,b,c C(b,d)⊗C(a,b) →∘ a,b,d C(a,d) ←∘ a,c,d C(c,d)⊗C(a,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 CC is associative;

and

C(b,b)⊗C(a,b) →∘ a,b,b C(a,b) ←∘ a,a,b C(a,b)⊗C(a,a) ↑ j b⊗Id C(a,b) ↗ l r↖ ↑ Id C(a,b)⊗j a I⊗C(a,b) C(a,b)⊗I \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.

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 VV is a monoidal category, then an alternative way of viewing a VV-category is as a set XX together with a (lax) monoidal functor Φ=Φ d\Phi = \Phi_d of the form

V op→yon VSet V→dSet X×XV^{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 XX, i.e., the local hom-category hom(X,X)\hom(X, X) in the bicategory of spans of sets. Given an VV-category (X,d:X×X→V)(X, d: X \times X \to V) under the ordinary definition, the corresponding monoidal functor Φ\Phi takes an object vv of V opV^{op} to the span

Φ(v) x,y:=hom V(v,d(x,y))\Phi(v)_{x, y} := \hom_V(v, d(x, y))

Under the composition law, we get a natural map

hom(v,d(x,y))×hom(v′,d(y,z))→hom(v⊗v′,d(x,y)⊗d(y,z))→hom(1,comp)hom(v⊗v′,d(x,z))\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 Φ(v)∘Φ(v′)→Φ(v⊗v′)\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 Φ=Φ d\Phi = \Phi_d induces an MM-enrichment on XX, and the two notions are equivalent.

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

Set V→Set X×XSet^V \to Set^{X \times X}

since bicontinuous functors of the form Set V→Set X×XSet^V \to Set^{X \times X} are precisely those of the form Set dSet^d for some function d:X×X→Vd: X \times X \to V, at least if VV is Cauchy complete.

Enrichment in a bicategory

Let BB be a bicategory, and write ⊗\otimes for horizontal (1-cell) composition (written in Leibniz order). A category enriched in the bicategory BB consists of a set XX together with

A function p:X→B 0p: X \to B_0,
A function hom:X×X→B 1\hom: X \times X \to B_1, satisfying the typing constraint hom(x,y):p(x)→p(y)\hom(x, y): p(x) \to p(y),
A function ∘:X×X×X→B 2\circ: X \times X \times X \to B_2, satisfying the constraint ∘ x,y,z:hom(y,z)⊗hom(x,y)→hom(x,z)\circ_{x, y, z}: \hom(y, z) \otimes \hom(x, y) \to \hom(x, z),
A function j:X→B 2j: X \to B_2, satisfying the constraint j x:1 p(x)→hom(x,x)j_x: 1_{p(x)} \to \hom(x, x),

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

If XX, YY are sets which come equipped with enrichments in BB, then a BB-functor consists of a function f:X→Yf: X \to Y such that p Y∘f=p Xp_Y \circ f = p_X, together with a function f 1:X×X→B 2f_1: X \times X \to B_2, satisfying the constraint f 1(x,y):hom X(x,y)→hom Y(f(x),f(y))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, jj of XX and YY. (Diagram to be inserted, perhaps.)

Categories enriched in bicategories were originally introduced by Bénabou under the name polyad. These can be seen as many-object generalisations of monads, since a category XX enriched in a bicategory BB is precisely a monad in BB when XX has one object.

Enrichment in a double category

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

for each x∈Xx\in X, an object p(x)p(x) of DD,
for each x,y∈Xx,y\in X, a horizontal arrow hom(x,y):p(x)→p(y)\hom(x, y)\colon p(x) \to p(y) in DD,
for each x,y,z∈Xx,y,z\in X, a 2-cell in DD:
p(x) →hom(x,y) p(y) →hom(y,z) p(z) ‖ ∘ x,y,z ‖ p(x) →hom(x,z) p(z)\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∈Xx\in X, a 2-cell in DD:
p(x) →id p(x) ‖ ‖ p(x) →hom(x,x) p(x)\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 DD; the vertical arrows of DD play no role in the definition. However, they do play a role when it comes to define functors between DD-enriched categories. Namely, if X\mathbf{X} and Y\mathbf{Y} are DD-enriched categories, then a DD-functor f:X→Yf\colon \mathbf{X}\to \mathbf{Y} consists of:

a function f:X→Yf\colon X\to Y,
for each x∈Xx\in X a vertical arrow f x:p X(x)→p Y(f(x))f_x\colon p_X(x) \to p_Y(f(x)) in DD,
for each x,y∈Xx,y\in X a 2-cell in DD:
p(x) →hom X(x,y) p(y) f x↓ ↓ f y p(f(x)) →hom Y(f(x),f(y)) p(f(y))\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 DD is vertically discrete, i.e. just a bicategory BB with no nonidentity vertical arrows, then this is just the same as a BB-functor as defined above. However, for many DD 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)), 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

Also called “change of base”, this typically refers to a monoidal functor between bases of enrichment V→WV \to W, inducing a 2-functor VV-Cat→WCat \to W-Cat, enabling constructions in VV-CatCat to be transferred to constructions in WW-Cat, as follows.

Passage between ordinary categories and enriched categories

Every KK-enriched category CC has an underlying ordinary category, usually denoted C 0C_0, defined by C 0(x,y)=K(I,hom(x,y))C_0(x,y) = K(I, hom(x,y)) where II is the unit object of KK.

If K(I,−):K→SetK(I, -): K \to Set has a left adjoint −⋅I:Set→K- \cdot I: Set \to K (taking a set SS to the tensor, aka the copower S⋅IS \cdot I, viz. the coproduct of an SS-indexed set of copies of II), then any ordinary category CC can be regarded as enriched in KK by forming the composite

Ob(C)×Ob(C)→homSet→−⋅IKOb(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 KK-Cat.

Lax monoidal functors

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

Moreover, this operation is itself functorial from MonCatMonCat to 2Cat2Cat. In particular, any monoidal adjunction K⇄LK\rightleftarrows L gives rise to a 2-adjunction KCat⇄LCatK Cat\rightleftarrows L Cat (and also for profunctors). The adjunction Cat⇄KCatCat \rightleftarrows K Cat described above is a special case of this arising from the adjunction −⋅I:Set⇄K:K(I,−)-\cdot I: Set \rightleftarrows K : K(I,-).

This and further properties of such “change of base” are explored in Crutwell 14

Tensor product of enriched categories

If VV is a symmetric monoidal category, then there is a tensor product of V-enriched categories (see enriched product category) which makes the category V Cat 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.

Enrichment versus internalization

See enrichment versus internalisation.

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 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 category enriched in Grpd is a strict (2,1)-category.
A strict n-category is a category enriched over strict (n−1)(n-1)-categories. In the limit n→∞n \to \infty this leads to strict omega-categories.
An algebroid, or linear category, is a category enriched over Vect. Here VectVect is the category of vector spaces over some fixed field KK, equipped with its usual tensor product. It is common to emphasize the dependence on KK and call a category enriched over Vect a KK-linear category.
More generally, if KK is any commutative ring, a category enriched over KK\,Mod is sometimes called a KK-linear category.
In particular, taking KK to be ℤ\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,∞],≥)([0, \infty], \geq) of extended positive real numbers, where ⊗\otimes is ++.
An ultrametric space is a category enriched over the poset ([0,∞],≥)([0, \infty], \geq) of extended positive real numbers, where ⊗\otimes is max\max.
A preorder 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 GG may be modeled by a category enriched over the discrete category on the set GG, 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 →Cat^{\to}, and similarly M-categories are categories enriched over a subcategory of Set →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 for details.
Lawvere theories can be represented as enriched categories as well; see Garner 2013 and Garner and Power 2017 for details.

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]
Jean-Marie Maranda, Formal categories, Canadian Journal of Mathematics 17 (1965) 758-801 [doi:10.4153/CJM-1965-076-0, 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]