nLab
closed category

Idea

A category $C$ is closed if for any pair $a,b$ of object the collection of morphisms from $a$ to $b$ can be regarded as forming itself an object of $C$.

This object is often denoted $\mathrm{hom}(a,b)$ or $[a,b]$ or similar and often addressed at the internal hom-object or simply the internal hom.

A familiar kind of closed categories are closed monoidal categories. However, there is also a definition of closed category that does not require the category to already be monoidal. A monoidal structure $\otimes$, if it exists, can then be universally characterized as a left adjoint to the internal-hom, dual to the above characterization of internal-homs as right adjoints to $\otimes$. See Eilenberg-Kelly, referenced below.

While this is less fashionable, in some cases it is more obvious what the correct internal-homs are than what the correct tensor product is, so the latter was originally defined as an adjoint to the former. This is the case for the Gray tensor product and was probably the case for abelian groups as well.

Definition

A closed category is a category $C$ together with the following data:

• A functor $[-,-]:C^{\mathrm{op}}\times C\to C$, called the internal hom-functor.
• An object $I\in C$ called the unit.
• A natural isomorphism $i:{\mathrm{Id}}_C\cong [I,-]$.
• A transformation $j_X:I\to [X,X]$, extranatural in $X$.
• A transformation $L_{YZ}^X:[Y,Z]\to [[X,Y],[X,Z]]$, natural in $Y$ and $Z$ and extranatural in $X$.

which is required to satisfy the following axioms.

• The map $\gamma :C(X,Y)\to C(I,[X,Y])$ defined by $f\mapsto [1,f](j_X)$ is a bijection.

• The following diagram commutes for any $X,Y$.

  $$\begin{array}{ccc}I& \stackrel{j_Y}{\to }& [Y,Y]\\ & {}_{j_{[X,Y]}}\searrow & {\downarrow }^{L_{YY}^X}\\ & & [[X,Y],[X,Y]]\end{array}$$\array{I & \overset{j_Y}{\to} & [Y,Y]\\& _{\mathllap{j_{[X,Y]}}}\searrow & \downarrow^{L^X_{Y Y}}\\ & & [[X,Y],[X,Y]]}

• The following diagram commutes for any $X,Y$.

  $$\begin{array}{ccc}[X,Y]& \stackrel{L_{XY}^X}{\to }& [[X,X],[X,Y]]\\ & {}_{i_{[X,Y]}}\searrow & {\downarrow }^{[j_X\mathrm{,1}]}\\ & & [I,[X,Y]]\end{array}$$\array{[X,Y] & \overset{L^X_{X Y}}{\to} & [[X,X],[X,Y]]\\& _{\mathllap{i_{[X,Y]}}}\searrow & \downarrow^{[j_X,1]}\\ & & [I,[X,Y]]}

• The following diagram commutes for any $Y,Z$.

  $$\begin{array}{ccc}[Y,Z]& \stackrel{L_{YZ}^I}{\to }& [[I,Y],[I,Z]]\\ & {}_{[1,i_Z]}\searrow & {\downarrow }^{[i_Y\mathrm{,1}]}\\ & & [Y,[I,Z]]\end{array}$$\array{[Y,Z] & \overset{L^I_{Y Z}}{\to} & [[I,Y],[I,Z]]\\& _{\mathllap{[1,i_Z]}}\searrow & \downarrow^{[i_Y,1]}\\ & & [Y,[I,Z]]}

• Finally, the following diagram commutes for any $X,Y,U,V$.

  $$\begin{array}{ccc}[U,V]& \stackrel{L_{UV}^Y}{\to }& [[Y,U],[Y,V]]\\ {}^{L_{UV}^X}\downarrow & & \\ [[X,U],[X,V]]& & {\downarrow }^{[1,L_{YV}^X]}\\ {}^{L_{[X,U],[X,V]}^{[X,Y]}}\downarrow & & \\ [[[X,Y],[X,U]],[[X,Y],[X,V]]]& \underset{[L_{YU}^X\mathrm{,1}]}{\to }& [[Y,U],[[X,Y],[X,V]]]\end{array}$$\array{[U,V] & \overset{L^Y_{U V}}{\to} & [[Y,U],[Y,V]]\\^{L^X_{U V}}\downarrow && \\ [[X,U],[X,V]] & & \downarrow^{[1,L^X_{Y V}]} \\ ^{L^{[X,Y]}_{[X,U],[X,V]}}\downarrow && \\ [[[X,Y],[X,U]],[[X,Y],[X,V]]] & \underset{[L^X_{Y U},1]}{\to} & [[Y,U],[[X,Y],[X,V]]]}

This definition is from Manzyuk’s paper below. It differs slightly from Eilenberg-Kelly’s original definition, which assumed in addition an “underlying-set-functor” $U:C\to \mathrm{Set}$ as part of the structure, with an axiom asserting that $U([X,Y])=C(X,Y)$ and that the resulting isomorphism

$$C(X,X)=U([X,X])\stackrel{U{i}_{[X,X]}}{\to }U([I,[X,X]])=C(I,[X,X])$$C(X,X) = U([X,X]) \overset{U i_{[X,X]}}{\to} U([I,[X,X]]) = C(I,[X,X])

sends $1_X$ to $j_X$. The two are essentially equivalent, and the one given here is perhaps a little simpler and avoids introducing superfluous data.

Examples

• Any closed monoidal category gives a closed category, by simply forgetting the tensor product and remembering only the internal-hom. Most examples seem to be of this sort, although as remarked above it is often the case that the closed structure is “primary” and the tensor product is defined as a left adjoint to it. Notice also, as discussed below that every closed category arises as the full subcategory of a closed monoidal category.

• Any multicategory which has a unit, i.e. an object $I$ such that $C(;Y)\cong C(I;Y)$ naturally, and is closed in the sense that for any $Y,Z$ there is an object $[Y,Z]$ with natural isomorphisms $C(X_1,\dots ,X_n,Y;Z)\cong C(X_1,\dots ,X_n;[Y,Z])$, gives rise to a closed category. Conversely, from any closed category we can construct a multicategory of this sort, by defining the multimaps as $C(X_1,\dots ,X_n;Z)=C(X_1,[X_2,\dots ,[X_n,Z]])$. Thus closed categories are essentially equivalent to closed unital multicategories.

Properties

Embedding into closed monoidal categories

By a result due to Miguel LaPlaza, every closed category embeds fully and faithfully into a closed monoidal category by a strong closed functor, i.e., one respecting closed structure up to suitably coherent isomorphism, and this closed functor is also strong monoidal if the original closed category is closed monoidal.

Monadicity and 2-categories

Since the notion of closed category involves a contravariant functor and extranatural transformations, it cannot be expected to be 2-monadic over the 2-category Cat. It is, however, 2-monadic over the 2-category ${\mathrm{Cat}}_g$ of categories, functors, and natural isomorphisms, the core of Cat. In this way we obtain a 2-category $\mathrm{ClCat}$ of closed categories, strong closed functors, and closed natural transformations. One can also define a notion of non-strong, or “lax,” closed functor; although these do not seemingly arise from the 2-monad in question, they generalize lax monoidal functors between closed monoidal categories.

References

Closed categories were first defined here:

• Samuel Eilenberg and Max Kelly, Closed categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965).

They were shown to be equivalent to closed unital multicategories here:

• Oleksandr Manzyuk, Closed categories vs. closed multicategories, arXiv.

You can get some of the idea from a post by Owel Biesel at the $n$-Café.

LaPlaza’s theorem on embedding closed categories in closed monoidal categories is given in

• Miguel LaPlaza, Embedding of Closed Categories Into Monoidal Closed Categories, Trans. AMS, Vol. 233 (1977).