nLab enriched category

Idea

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

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

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, $K$-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

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

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

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.

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 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

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

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.

Enrichment in a bicategory

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

• 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 $\mathbf{B} M$; 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 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

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)), 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 $V \to W$, 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 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 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 Crutwell 14

Tensor product of enriched categories

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 $V Cat$ 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 $(n-1)$-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 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 $G$ 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 Garner 2018 for details.

• Lawvere theories can be represented as enriched categories as well; see Garner 2013 and Garner and Power 2017 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]

• 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]

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:

• 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, pdf)

Textbook accounts:

• 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, tac: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)

• Emily Riehl, Basic concepts of enriched category theory, chapter 3 in: Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)

• Niles Johnson, Donald Yau, Section 1.3 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

Also:

• Francis Borceux, I. Stubbe, Short introduction to enriched categories (pdf)

With an eye towards application in homotopy theory:

• Birgit Richter, Section 9 of: From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (doi:10.1017/9781108855891, book webpage, pdf)

Discussion of change of enriching category is in

• Geoff Cruttwell, chapter 4 of Normed spaces and the Change of Base for Enriched Categories, 2014 (pdf)

Vista of some modern generalizations is in

• Tom Leinster, Generalized enrichment for categories and multicategories, math.CT/9901139

• Tom Leinster, Generalized enrichment of categories, Journal of Pure and Applied Algebra 168 (2002), no. 2-3, 391-406, math.CT/0204279

• John Armstrong, Enriched categories

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:1707.08694

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

On enrichment in more general structures:

• Christian Lair, Systèmes tensoriels et systèmes enrichis, Diagrammes 43 (2000): 3-48.