User:Fropuff/Drafts/Closed category

Draft in progress

In category theory, a branch of mathematics, a closed category is a category which possess a functor, called the internal Hom functor that behaves very much like the ordinary (external) Hom functor except that it takes values in the same category, rather than the category of sets.

For example, in the category of abelian groups, Ab, the set of all group homomorphisms between two abelian groups can be given the structure of an abelian group in a natural way, so the ordinary Hom functor can be taken to have values in Ab and not just Set. For many closed concrete categories, $$C$$, the internal Hom can be obtained by adding additional "structure" to the ordinary Hom set, so that the forgetful functor $$\mathcal C\to\mathbf{Set}$$ takes the internal Hom onto the external Hom. However, this need not be the case&mdash;sometimes the internal Hom functor takes on quite a different form than the external Hom functor. What remains true is that every closed category $$\mathcal C$$ comes equipped with a (not necessarily faithful) functor $$V\colon \mathcal C\to\mathbf{Set}$$ that maps the internal Hom onto the external Hom. For concrete closed categories this functor may, or may not, coincide with the forgetful functor.

A rich collection of examples is provided by the class of closed monoidal categories, where the internal Hom functor forms an adjoint to the monoidal product $$\otimes$$. Most examples are of this form, but sometimes it is more natural to start with the closed structure and define the monoidal product as adjoint to the internal Hom, rather than the other way around. It has been shown that every closed category can be embedded as a full subcategory of a closed monoidal category.

Definition
A closed category is a category $$\mathcal{C}$$ together with the following data: which satisfy the following five axioms:
 * a bifunctor $$[-, -]\colon \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}$$ called the internal Hom functor,
 * an object $$I$$ of $$\mathcal{C}$$ called the unit object,
 * a natural isomorphism $$i_X\colon X \to [I, X]$$,
 * an extranatural transformation $$j_X\colon I \to [X, X]$$,
 * a transformation $$L^X_{YZ}\colon [Y,Z]\to X,Y],[X,Z$$ natural in $$Y$$ and $$Z$$, and extranatural in $$X$$.
 * 1) the map $$\gamma_{XY}\colon\operatorname{Hom}(X, Y)\to \operatorname{Hom}(I, [X,Y])$$ given by $$\gamma_{XY}(f) = [X, f]\circ j_X = [f, Y]\circ j_Y$$ is a bijection.
 * 2) $$j_{[X,Y]} = L^X_{YY}\circ j_Y$$,
 * 3) $$i_{[X,Y]} = \left[j_X, [X,Y]\right]\circ L^X_{XY}$$,
 * 4) $$\left[X,i_Y\right] = \left[i_X,[I,Y]\right]\circ L^I_{XY}$$,
 * 5) $$\left[[Y,U],L^X_{YV}\right]\circ L^Y_{UV} = \left[L^X_{YU},\left[[X,Y],[X,V]\right]\right]\circ L^{[X,Y]}_{[X,U][X,V]}\circ L^X_{UV}$$,

The first axiom says that the functor $$V\colon\mathcal C\to\mathbf{Set}$$ given by
 * $$V = \operatorname{Hom}(I,-)$$

maps the internal Hom object $$[X, Y]$$ onto a set isomorphic to the external Hom set $$\operatorname{Hom}(X,Y)$$:
 * $$V([X,Y])\cong\operatorname{Hom}(X,Y)$$

In other words, the bifunctor $$V[-,-]$$ is naturally isomorphic to the external Hom functor $$\operatorname{Hom}(-,-)$$.

Examples

 * Cartesian closed categories are closed categories. In particular, any topos is closed. The canonical example is the category of sets.
 * Compact closed categories are closed categories. The canonical example is the category FdVect with finite-dimensional vector spaces as objects and linear maps as morphisms.
 * More generally, any monoidal closed category is a closed category. In this case, the object $$I$$ is the monoidal unit.

Properties
In any closed category the following statements hold:
 * $$i_{[I,X]} = [I,i_X] : [I, X]\to\left[I,[I,X]\right]$$. This follows purely from the fact that $$i\colon 1_{\mathcal C}\to [I,-]$$ is a natural isomorphism.
 * $$\gamma_{XX}(\operatorname{id}_X) = j_X$$
 * $$i_I = j_I : I\to[I,I]$$
 * $$\gamma_{IX} = \operatorname{Hom}(I,i_X) : \operatorname{Hom}(I, X)\to\operatorname{Hom}(I,[I,X])$$
 * the endomorphism monoid of $$I$$ is commutative