View source for 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 functor|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 of a family of objects in a category|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 [[Modules, category of|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> M. Bunge, ''Matematika'' , '''16''' : 2 (1972) pp. 11–46 {{MR|0360082}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> F.W. Lawvere, "Introduction" F.W. Lawvere (ed.) , ''Toposes, algebraic geometry and logic (Dalhousic Univ., Jan. 1971)'' , ''Lect. notes in math.'' , '''274''' , Springer (1972) {{MR|0376798}} {{ZBL|0249.18015}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> E.J. Dubuc, "Kan extensions in enriched category theory" , Springer (1970) {{MR|0280560}} {{ZBL|0228.18002}} </TD></TR>
</table>



====Comments====


====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 {{MR|}} {{ZBL|0232.18001}} </TD></TR>
</table>

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