enriched category



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 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,yob(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. 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.

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.

Enrichment in a monoidal category

Let VV be a monoidal category with

  • tensor product :V×VV\otimes : V \times V \to V;

  • tensor unit IObj(V)I \in Obj(V);

  • associator α a,b,c:(ab)ca(bc)\alpha_{a,b,c} : (a \otimes b)\otimes c \to a \otimes (b \otimes c);

  • left unitor l a:Iaal_a : I \otimes a \to a;

  • right unitor r a:aIar_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 aObj(C)a \in Obj(C) a morphism j a:IC(a,a)j_a : I \to C(a,a) – called the identity element

  • such the following diagrams commute:

for all a,b,c,dObj(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,dId 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;


C(b,b)C(a,b) a,b,b C(a,b) a,a,b C(a,b)C(a,a) j bId C(a,b) l r Id C(a,b)j a IC(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.

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 opyon VSet VdSet 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×XV)(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(vv,d(x,y)d(y,z))hom(1,comp)hom(vv,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(vv)\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 VSet X×XSet^V \to Set^{X \times X}

since bicontinuous functors of the form Set VSet X×XSet^V \to Set^{X \times X} are precisely those of the form Set dSet^d for some function d:X×XVd: 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:XB 0p: X \to B_0,
  • A function hom:X×XB 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×XB 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:XB 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. Viewing a monoidal category MM as a 1-object bicategory ΣM\Sigma M, the notion of enrichment in MM coincides with the notion of enrichment in the bicategory ΣM\Sigma M.

If XX, YY are sets which come equipped with enrichments in BB, then a BB-functor consists of a function f:XYf: X \to Y such that p Yf=p Xp_Y \circ f = p_X, together with a function f 1:X×XB 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.)

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 xXx\in X, an object p(x)p(x) of DD,
  • for each x,yXx,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,zXx,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 xXx\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:XYf\colon \mathbf{X}\to \mathbf{Y} consists of:

  • a function f:XYf\colon X\to Y,
  • for each xXx\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,yXx,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.

Change of enriching category

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,):KSetK(I, -): K \to Set has a left adjoint I:SetK- \cdot I: Set \to K (taking a set SS to the tensor or copower SIS \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)homSetIKOb(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:KLF: 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 KLK\rightleftarrows L gives rise to a 2-adjunction KCatLCatK Cat\rightleftarrows L Cat (and also for profunctors). The adjunction CatKCatCat \rightleftarrows K Cat described above is a special case of this arising from the adjunction I:SetK:K(I,)-\cdot I: Set \rightleftarrows K : K(I,-).

This and further properties of such “change of base” are explored in Geoff Cruttwell’s thesis.

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 KK-enriched category, the objects still form a set (or a proper class) while the arrows are replaced by objects of KK, while in a category internal to KK, both the set of objects and the set of arrows are replaced by objects of KK.

Another difference is that for KK-enriched categories, KK can be any monoidal category, while for KK-internal categories, it must have pullbacks, which can be thought of as a generalization of cartesian monoidal structure. In particular, a KK-internal category with one object (that is, whose object-of-objects is a terminal object) is a monoid in KK with respect to the cartesian product, whereas a one-object KK-enriched category is a monoid in KK 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) and so are both forms of generalized multicategory. On the other hand, when KK is an \infty-extensive category, such as Set or simplicial sets (or more generally any Grothendieck topos), (small) KK-enriched categories can be identified with KK-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 standard reference on enriched categories is

  • Max Kelly, Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp.; remake: TAC reprints 10, tac,pdf


Vista of some modern generalizations is in

Change of base is discussed in

Further examples are discussed in

  • Richard Garner, Lawvere theories, finitary monads and Cauchy-completion, Journal of Pure and Applied Algebra, 2014, arxiv

  • Richard Garner and John Power, An enriched view on the extended finitary monad–Lawvere theory correspondence, 2017, arxiv

  • Richard Garner, An embedding theorem for tangent categories, Advances in Mathematics, 2018, doi, arxiv

Revised on January 17, 2018 00:44:18 by Mike Shulman (