Closed category

A category with an additional structure, thanks to which the internal Hom-functor can be used as a right-adjoint functor to the abstract tensor product.

A category $\mathfrak{M}$ is said to be closed if a bifunctor $\otimes: \mathfrak{M} \times \mathfrak{M} \rightarrow \mathfrak{M}$ (see Functor) and a distinguished object $I$ are given on it, and if it admits natural isomorphisms $$ \alpha_{ABC} : (A \otimes B) \otimes C \rightarrow A \otimes (B \otimes C)\ \ \ \text{associativity,} $$ $$ \lambda_A : I \otimes A \rightarrow A\ \ \ \text{left identity,} $$ $$ \rho_A : A \otimes I \rightarrow A\ \ \ \text{right identity,} $$ $$ \kappa_{AB} : A \otimes B \rightarrow B \otimes A\ \ \ \text{commutativity,} $$ such that the following conditions are satisfied: 1) the natural isomorphisms $\alpha, \lambda, \rho, \kappa$ are coherent; and 2) every functor $$ H_{AB}(X) = H_{\mathfrak{M}}(A\otimes X,B) : \mathfrak{M} \rightarrow \mathsf{Set} $$ where $\mathsf{Set}$ is the category of sets, is representable. The representing objects are usually denoted by $\mathrm{Hom}_{\mathfrak{M}}(A,B)$, and they can be regarded as the values of the bifunctor $\mathrm{Hom}_{\mathfrak{M}}:\mathfrak{M}^* \times \mathfrak{M} \rightarrow \mathfrak{M}$ (the internal Hom-functor) on objects. If the bifunctor $\otimes$ coincides with a product and $I$ is a right zero (terminal object) of $\mathfrak{M}$, then $\mathfrak{M}$ is called a Cartesian-closed category.

The following categories are Cartesian closed: the category of sets, the category of small categories (cf. Small category) and the category of sheaves of sets over a topological space. The following categories are closed: the category of modules over a commutative ring with an identity and the category of real (or complex) Banach spaces and linear mappings with norm not exceeding one.

References

[1]	M. Bunge, Matematika , 16 : 2 (1972) pp. 11–46 MR0360082
[2]	F.W. Lawvere, "Introduction" F.W. Lawvere (ed.) , Toposes, algebraic geometry and logic (Dalhousic Univ., Jan. 1971) , Lect. notes in math. , 274 , Springer (1972) MR0376798 Zbl 0249.18015
[3]	E.J. Dubuc, "Kan extensions in enriched category theory" , Springer (1970) MR0280560 Zbl 0228.18002

References

[a1]	S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 Zbl 0232.18001