A category $C$ is closed if for any pair $a, b$ of objects the collection of morphisms from $a$ to $b$ can be regarded as forming itself an object of $C$.
This object is often denoted $hom(a,b)$ or $[a,b]$ or similar and often addressed as 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 an (enriched) left adjoint to the internal-hom, dual to the above characterization of internal-homs as right adjoints to $\otimes$; see below.
Although it may seem that in general the closed structure is less intuitive to work with than the monoidal structure, in some cases it is in fact 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 the projective tensor product, for example, and was probably the case for abelian groups (the original notion of tensor product) as well.
A closed category is a category $C$ together with the following data:
A functor $[-,-] : C^{op} \times C \to C$, called the internal hom-functor.
An object $I\in C$ called the unit object.
A natural isomorphism $i\colon Id_C \cong [I,-]$.
A transformation $j_X\colon I \to [X,X]$, extranatural in $X$.
A transformation $L^X_{Y Z} \colon [Y,Z] \to [ [X,Y],[X,Z]]$, natural in $Y$ and $Z$ and extranatural in $X$.
which is required to satisfy the following axioms.
The following diagram commutes for any $X,Y$.
The following diagram commutes for any $X,Y$.
The following diagram commutes for any $Y,Z$.
The following diagram commutes for any $X,Y,U,V$.
Finally, the map $\gamma\colon C(X,Y) \to C(I,[X,Y])$ defined by $f \mapsto [1,f](j_X)$ is a bijection.
Note that both a left-closed (non-symmetric) monoidal category and a right-closed (non-symmetric) monoidal category give rise to a closed category in this sense. See the discussion on the nForum.
The above definition is the one used in LaPlaza and Manzyuk. It differs slightly from Eilenberg-Kelly’s original definition, which omitted the axiom involving $\gamma$ but assumed an “underlying-set-functor” $U\colon C \to Set$ as part of the structure, with an axiom asserting that $U([X,Y]) = C(X,Y)$ (naturally, i.e. as a strict equality of functors $C^{op}\times C\to Set$) and that the resulting isomorphism
sends $1_X$ to $j_X$. (This is then essentially a definition of $j_X$, so that it could be omitted from the definition.)
The two are essentially equivalent, and the one given here is perhaps a little simpler, as well as obeying the principle of equivalence. To be precise, every closed category in the sense of Eilenberg-Kelly is also one in the above sense, and conversely every closed category in the above sense is isomorphic to one in the sense of Eilenberg-Kelly. The latter has the same objects and internal-homs, but new hom-sets $\hat{C}(X,Y) \coloneqq C(I,[X,Y])$, and underlying-set functor $U(X) \coloneqq C(I,X)$ (note that this is different from $\hat{C}(I,X) = C(I,[I,X])$). See Manzyuk for the proof.
The fact (discussed below) that closed categories, in the above sense, are equivalent to closed multicategories with unit, suggests that there ought to be a version of closed categories that don’t necessarily have a unit object $I$. In fact in this case the definition becomes simpler because we can omit all the $i$‘s and $j$’s, but there is an additional wrinkle that we have to include “left evaluation” as a new operation. Furthermore, it seems that we have to use the Eilenberg-Kelly version of the definition, not the LaPlaza one; the “underlying set” functor $U:C\to Set$ corresponds to the nullary homsets $C(;-)$ of a multicategory.
A prounital closed category is a category $C$ together with the following data:
A functor $[-,-] : C^{op} \times C \to C$, called the internal hom-functor.
A functor $U : C \to Set$.
A transformation $L^X_{Y Z} \colon [Y,Z] \to [ [X,Y],[X,Z]]$, natural in $Y$ and $Z$ and extranatural in $X$.
A transformation $ev : U(X) \to C([X,Y], Y)$, natural in $X$ and extranatural in $Y$.
which is required to satisfy the following axioms.
$U([X,Y]) = C(X,Y)$ strictly and naturally, i.e. $U \circ [-,-] = C(-,-)$ as functors $C^{op}\times C\to Set$. (Note that this can be phrased in accord with the principle of equivalence by considering $U : C \to Set$ as a displayed category over $Set$.)
For any $x\in U(X)$ and $f\in C(X,Y) = U([X,Y])$ we have $U(ev_x)(f) = U(f)(x)$ in $U(Y)$.
For any $X,Y\in C$ we have $U(L^X_{Y Y})(1_Y) = 1_{[X,Y]}$. (Note that $L^X_{Y Y} : [X,X] \to [ [X,Y],[X,Y]]$, so $U(L^X_{Y Y})(1_Y) : C(X,X) \to C([X,Y],[X,Y])$.)
For any $X,Y\in C$ the composite $[X,Y] \xrightarrow{L^X_{X Y}} [ [X,X],[X,Y]] \xrightarrow{ev_{1_X}} [X,Y]$ is the identity.
The following diagram commutes for any $X,Y,U,V$.
Some axiom(s) relating $L$ to $ev$. For instance, for any $x\in U(X)$ the following diagram should commute:
The third and fourth axioms, plus naturality of $ev$, should imply a more general form of the third axiom: for any $f:X\to Y$, the composite $[Y,Z] \xrightarrow{L^X_{Y Z}} [ [X,Y],[X,Z]] \xrightarrow{ev_f} [X,Z]$ equals the functorial action $[f,Z]$.
I haven’t worked out exactly what axioms are required here.
A symmetric closed category is additionally equipped with natural isomorphisms $C^{X Y}_Z : [X,[Y,Z]] \cong [Y,[X,Z]]$ satisfying appropriate axioms; see deSchipper75 and DayLaPlaza78. These should correspond to symmetric multicategories.
Similarly, a “cartesian closed category” (there is an unfortunate terminological clash with cartesian closed category) should furthermore have morphisms $K^X_Y : Y\to [X,Y]$ and $T^X_Y : [X,[X,Y]]\to [X,Y]$ satisfying appropriate axioms. They should correspond to cartesian multicategories. The relation to combinatory logic suggests that it should also be possible to axiomatize the cartesian case by replacing $L$, $C$, and $T$ with a single family of morphisms $S^X_{Y Z} : [X,[Y,Z]]\to [ [X,Y],[X,Z]]$.
I don’t know of anywhere that the details are worked out for the cartesian case.
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 (in fact, as discussed below, every closed category arises as the full subcategory of a closed monoidal category). However, 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 (see below). For instance:
In the category Ab of abelian groups the internal-hom $[X,Y]$ is the set of group homomorphisms $X\to Y$ with the pointwise addition, giving a closed category.
More generally, for any commutative theory $T$, the category of $T$-algebras has an internal-hom $[X,Y]$ that is the set of $T$-algebra maps with the pointwise operations (this is again a $T$-algebra by commutativity).
In the category 2Cat of strict 2-categories, the internal-hom $[X,Y]$ can be taken to be the category of strict 2-functors $X\to Y$ and pseudo, lax, or oplax natural transformations between them, giving three closed categories.
In the category $Unif$ of uniform spaces the internal-hom $[X,Y]$ can be taken to be the set of uniformly continuous functions $X\to Y$ with either the uniformity of uniform convergence or of pointwise convergence, giving two closed categories.
In all these examples there is a corresponding tensor product, but its construction is much less intuitive, being essentially defined by a sort of adjoint functor theorem or initial lift that gives little information about what it actually looks like.
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 $C(X_1,\dots,X_n; Z) = C(I, [X_1,\dots,[X_n,Z]])$. Thus closed categories are essentially equivalent to closed unital multicategories.
The closed structure on a category $\mathcal{V}_0$ is an enrichment context $\mathcal{V}$ relative to which we can define a category of V-enriched categories. From this point of view, a closed structure is more natural than a monoidal structure since most (if not all) of this structure is forced if one insists for an enrichment context $\mathcal{V}$ for which $\mathcal{V}_0$ is self-enriched, that is, isomorphic to the underlying category $\mathcal{V}^e_0$ of some $\mathcal{V}$-enriched category $\mathcal{V}^e$. See category of V-enriched categories for details.
Suppose $C$ is a closed category such that the functor $[A,[B,-]] : C \to C$ is representable as a $C$-enriched functor. This means we have an object $A\otimes B$ and a $C$-natural isomorphism
Then $C$ is a (closed) monoidal category. See Eilenberg–Kelly (1965) for details.
Note that although this is analogous to the fact that a monoidal category with right adjoints to tensor product is a closed category, the requisite hypothesis is stronger. When $C$ is given as monoidal, we need only $Set$-natural isomorphisms of hom-sets $C(A\otimes B,X) \cong C(A,[B,X])$ to make it closed, whereas when $C$ is closed we need the above $C$-natural isomorphism of internal hom-objects to make it monoidal.
However, in the case when $C$ is a symmetric closed category, this stronger condition is unnecessary: to make $C$ (symmetric) monoidal it suffices to have external natural isomorphisms of hom-sets
Put differently, any symmetric closed category has an underlying (symmetric) promonoidal category, whereas for a non-symmetric closed category the associativity of the promonoidal structure is an extra condition. See Day-Laplaza, Proposition 2.3.
By a result due to Miguel LaPlaza (1977), 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. In particular, this functor is defined as the composition $q = y \circ p$ of the functor
sending an object $X \in C$ to the object (endofunctor) $[X,-] \in E = [C,C]$, together with the Yoneda embedding $y : E^\op \to [E, Set]$. The closed monoidal structure on the presheaf category $[E,Set]$ is given by Day convolution, using the monoidal structure of the category of endofunctors $E$ (see closed monoidal structure on presheaves over a monoidal category).
An alternative embedding was also sketched by Day (1974), and later studied by Day and LaPlaza (1978) in the case of symmetric closed categories. This embedding is based on the fact that the construction of a closed monoidal category of presheaves does not actually require the base category to be monoidal: it suffices for it to be a promonoidal category, and any closed category gives rise to such a promonoidal structure (perhaps a “lax” one with noninvertible associativity). So, instead of using the category of copresheaves on the monoidal category of endofunctors $[ [C,C],Set]$ as in (LaPlaza 1977), the embedding in (Day 1974) and (Day and LaPlaza 1978) uses $[C,Set]$ itself. One can then restrict to a small subcategory $A$ of $[C,Set]$ and compose further with a restricted version of its Yoneda embedding in order to obtain an embedding of $C$ into a closed monoidal category that also preserves certain colimits in $C$.
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 $Cat_g$ of categories, functors, and natural isomorphisms, the core of Cat. In this way we obtain a 2-category $ClCat$ 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.
There is a notion that is related to a bicategory in the same way that a closed category is related to a monoidal category, i.e. a horizontal categorification of a closed category. It has a set of “objects”, for any two objects $A,B$ a category $\mathcal{B}(A,B)$, and internal-hom functors like those in a closed bicategory. It was defined by Street under the name extension system, although our page extension system is about an unrelated concept with the same name.
Closed categories were first defined here:
Their coherence theorem was considered in terms of Kelly-Mac Lane graphs in
They were shown to be equivalent to closed unital multicategories here:
You can get some of the idea from a post by Owen Biesel at the $n$-Café.
LaPlaza’s theorem on embedding closed categories in closed monoidal categories is given in
The alternative embedding of closed categories based on promonoidal Day convolution is discussed in
Brian Day, An embedding theorem for closed categories, Lecture Notes in Mathematics 420 (1974), 55-64.
B. J. Day and M. L. Laplaza, On Embedding Closed Categories, Bull. Austral. Math. Soc. 18 (1978), 357-371.
The latter uses the notion of symmetric closed category, which was first defined in
The generalization to a bicategory-like notion (“extension systems”) appears in
Last revised on September 28, 2022 at 00:04:41. See the history of this page for a list of all contributions to it.