2.1.7 Enriched Category Theory
Let $\operatorname{\mathcal{C}}$ be a category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ denote the set of morphisms from $X$ to $Y$ in $\operatorname{\mathcal{C}}$. In many cases of interest, the sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be endowed with additional structure, which are respected by the composition law on $\operatorname{\mathcal{C}}$. To give a systematic discussion of this phenomenon, it is convenient to use the formalism of enriched category theory.
Definition 2.1.7.1. Let $\operatorname{\mathcal{A}}$ be a monoidal category with unit object $\mathbf{1}$. An $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ consists of the following data:
- $(1)$
A collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}})$.
- $(2)$
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, an object $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ of the monoidal category $\operatorname{\mathcal{A}}$.
- $(3)$
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism
\[ c_{Z,Y,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \]
in the category $\operatorname{\mathcal{A}}$, which we will refer to as the composition law.
- $(4)$
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism $e_{X}: \mathbf{1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ in the category $\operatorname{\mathcal{A}}$, which we refer to as the identity of $X$.
These data are required to satisfy the following conditions:
- $(A)$
For every quadruple of objects $W,X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the diagram
\[ \xymatrix@R =50pt@C=40pt{ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,Y) \ar [dd]^-{ c_{Z,Y,W} } \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes ( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) ) \ar [dd]^-{\alpha } \ar [ur]^{ \operatorname{id}\otimes c_{Y,X,W} } & \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,Z ) \\ ( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y) ) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) \ar [dr]_{ c_{Z,Y,X} \otimes \operatorname{id}} & \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) \ar [uu]_-{ c_{Z,X,W} } } \]
commutes. Here $\alpha $ denotes the associativity constraint on the monoidal category $\operatorname{\mathcal{A}}$.
- $(U)$
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the diagrams
\[ \xymatrix@C =50pt{ \mathbf{1} \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [rr]^-{e_{Y} \otimes \operatorname{id}} \ar [dr]_-{\lambda } & & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Y) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [dl]^-{ c_{Y,Y,X} } \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) & } \]
\[ \xymatrix@C =50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \otimes \mathbf{1} \ar [rr]^-{\operatorname{id}\otimes e_ X} \ar [dr]_-{\rho } & & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \ar [dl]^-{ c_{Y,X,X} } \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) & } \]
commute, where $\lambda $ and $\rho $ denote the left and right unit constraints on $\operatorname{\mathcal{A}}$ (see Construction 2.1.2.17).
Example 2.1.7.2 (Categories Enriched Over Sets). Let $\operatorname{\mathcal{A}}= \operatorname{Set}$ be the category of sets, endowed with the monoidal structure given by the cartesian product (see Example 2.1.3.2). Then an $\operatorname{\mathcal{A}}$-enriched category (in the sense of Definition 2.1.7.1) can be identified with a category in the usual sense.
Example 2.1.7.3. Let $\operatorname{\mathcal{A}}$ be a monoidal category. If $\operatorname{\mathcal{C}}$ is a category enriched over $\operatorname{\mathcal{A}}$ and $X$ is an object of $\operatorname{\mathcal{C}}$, then the composition law
\[ c_{X,X,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \]
exhibits $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ as an algebra object of $\operatorname{\mathcal{A}}$, in the sense of Example 2.1.5.21. Moreover, this construction induces a bijection
\[ \{ \text{$\operatorname{\mathcal{A}}$-Enriched Categories $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X \} $} \} \simeq \{ \text{Algebra objects of $\operatorname{\mathcal{A}}$} \} . \]
Consequently, the theory of enriched categories can be regarded as a generalization of the theory of associative algebras (See Example 2.1.7.14 for a more precise statement).
Example 2.1.7.5 (The Underlying Category of an Enriched Category). Let $\operatorname{\mathcal{A}}$ be a monoidal category and let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{Set}$ be the functor given by $F( A ) = \operatorname{Hom}_{\operatorname{\mathcal{A}}}( \mathbf{1}, A)$, endowed with the lax monoidal structure of Example 2.1.5.16. If $\operatorname{\mathcal{C}}$ is a category enriched over $\operatorname{\mathcal{A}}$, then we can apply the construction of Remark 2.1.7.4 to obtain a $\operatorname{Set}$-enriched category, which we can identify with an ordinary category (Example 2.1.7.2). We will refer to this category as the underlying category of the $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$, and we will generally abuse notation by denoting it also by $\operatorname{\mathcal{C}}$. Concretely, this underlying category has the same objects as the enriched category $\operatorname{\mathcal{C}}$, with morphism sets given by the formula $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{A}}}( \mathbf{1}, \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$.
Example 2.1.7.7 (Enrichment in Vector Spaces). Let $k$ be a field and let $\operatorname{Vect}_{k}$ denote the category of vector spaces over $k$, endowed with the monoidal structure given by tensor product over $k$ (Example 2.1.3.1). Then choosing an $\operatorname{Vect}_{k}$-enrichment of $\operatorname{\mathcal{C}}$ is equivalent to endowing each of the sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ with the structure of a $k$-vector space, for which the composition maps
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z) \]
are $k$-bilinear.
Example 2.1.7.8 (Topologically Enriched Categories). Let $\operatorname{Top}$ denote the category of topological spaces, endowed with the monoidal structure given by the cartesian product (Example 2.1.3.2). We will refer to a $\operatorname{Top}$-enriched category as a topologically enriched category. Note that the functor $F$ of Example 2.1.7.5 is (canonically isomorphic to) the forgetful functor $\operatorname{Top}\rightarrow \operatorname{Set}$. Consequently, if $\operatorname{\mathcal{C}}$ is a topologically enriched category, then the underlying ordinary category $\operatorname{\mathcal{C}}_0$ can be described concretely as follows:
The objects of the ordinary category $\operatorname{\mathcal{C}}_0$ are the objects of the $\operatorname{Top}$-enriched category $\operatorname{\mathcal{C}}$.
Given a pair of objects $X,Y \in \operatorname{\mathcal{C}}_0$, a morphism $f$ from $X$ to $Y$ (in the ordinary category $\operatorname{\mathcal{C}}_0$) is a point of the topological space $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y)$.
Given a pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}_0$, the composition $g \circ f$ is given by the image of $(g,f)$ under the continuous map
\[ c_{Z,Y,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Z ). \]
It follows that, for any ordinary category $\operatorname{\mathcal{C}}_0$, promoting $\operatorname{\mathcal{C}}_0$ to a topologically enriched category $\operatorname{\mathcal{C}}$ is equivalent to endowing each of the morphism sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( X, Y )$ with a topology for which the composition maps $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( Y, Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( X,Y ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}(X, Z)$ are continuous.
Exercise 2.1.7.9 (Uniqueness of Identities). Let $\operatorname{\mathcal{A}}$ be a monoidal category. A nonunital $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ consists of a collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$ of objects of $\operatorname{\mathcal{C}}$, together with objects $\{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \} _{X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})}$ of the category $\operatorname{\mathcal{A}}$ and composition laws
\[ c_{Z,Y,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \]
which satisfy the associative law $(A)$ appearing in Definition 2.1.7.1. Show that, if a nonunital $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ can be promoted to an $\operatorname{\mathcal{A}}$-enriched category $\overline{\operatorname{\mathcal{C}}}$, then $\overline{\operatorname{\mathcal{C}}}$ is unique: that is, the identity maps $e_{X}: \mathbf{1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ are determined by axiom $(U)$ of Definition 2.1.7.1.
Definition 2.1.7.10. Let $\operatorname{\mathcal{A}}$ be a monoidal category, and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\operatorname{\mathcal{A}}$-enriched categories. An $\operatorname{\mathcal{A}}$-enriched functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ consists of the following data:
- $(1)$
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and object $F(X) \in \operatorname{Ob}(\operatorname{\mathcal{D}})$.
- $(2)$
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism
\[ F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
in the category $\operatorname{\mathcal{A}}$.
These data are required to satisfy the following conditions:
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the morphism $e_{F(X)}: \mathbf{1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(X) )$ factors as a composition
\[ \mathbf{1} \xrightarrow { e_{X} } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, X) \xrightarrow { F_{X,X} } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(X) ). \]
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r] \ar [d]^{ F_{Y,Z} \otimes F_{X,Y} } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{ F_{X,Z} } \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) ) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \ar [r] & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) } \]
commutes (in the category $\operatorname{\mathcal{A}}$); here the horizontal maps are given by the composition laws on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.
Notation 2.1.7.11 (The Category of Enriched Categories). Let $\operatorname{\mathcal{A}}$ be a monoidal category. We say that an $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ is small if the collection of objects $\operatorname{Ob}(\operatorname{\mathcal{C}})$ is small. The collection of small $\operatorname{\mathcal{A}}$-enriched categories can itself be organized into a category $\operatorname{Cat}(\operatorname{\mathcal{A}})$, whose morphisms are given by $\operatorname{\mathcal{A}}$-enriched functors (in the sense of Definition 2.1.7.10).
Example 2.1.7.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be small categories, which we regard as $\operatorname{Set}$-enriched categories by means of Example 2.1.7.2. Then $\operatorname{Set}$-enriched functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.1.7.10) can be identified with functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in the usual sense. This identification determines an isomorphism of categories $\operatorname{Cat}\simeq \operatorname{Cat}(\operatorname{Set})$.
Example 2.1.7.14. Let $\operatorname{\mathcal{A}}$ be a monoidal category, let $A$ be an algebra object of $\operatorname{\mathcal{A}}$, which we can identify with an $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}_{A}$ having a single object $X$ (Example 2.1.7.3). For any $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{D}}$ containing an object $Y$, we have a canonical bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{$\operatorname{\mathcal{A}}$-Enriched Functors $F: \operatorname{\mathcal{C}}_{A} \rightarrow \operatorname{\mathcal{D}}$ with $F(X) = Y$} \} \ar [d]^{\sim } \\ \{ \text{Algebra homomorphisms $A \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(Y,Y)$} \} . } \]
In particular, if $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{B}$ for some other algebra object $B \in \operatorname{Alg}(\operatorname{\mathcal{A}})$, we obtain a bijection
\[ \operatorname{Hom}_{ \operatorname{Cat}(\operatorname{\mathcal{A}}) }( \operatorname{\mathcal{C}}_{A}, \operatorname{\mathcal{C}}_{B} ) \simeq \operatorname{Hom}_{ \operatorname{Alg}(\operatorname{\mathcal{A}})}( A, B). \]
In other words, the construction $A \mapsto \operatorname{\mathcal{C}}_{A}$ induces a fully faithful embedding $\operatorname{Alg}(\operatorname{\mathcal{A}}) \rightarrow \operatorname{Cat}(\operatorname{\mathcal{A}})$, whose essential image is spanned by those $\operatorname{\mathcal{A}}$-enriched categories having a single object.