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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225101.png" /> is said to be closed if a bifunctor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225102.png" /> (see [[Functor|Functor]]) and a distinguished object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225103.png" /> have been given on it, and if it admits natural isomorphisms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225104.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225105.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225106.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225107.png" /></td> </tr></table>

such that the following conditions are satisfied: 1) the natural isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225108.png" /> are coherent; and 2) every functor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c0225109.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c02251010.png" /> is the category of sets, is representable. The representing objects are usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c02251011.png" />, and they can be regarded as the values of the bifunctor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c02251012.png" /> (the internal Hom-functor) on objects. If the bifunctor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c02251013.png" /> coincides with a product and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c02251014.png" /> is a right zero (terminal object) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c02251015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022510/c02251016.png" /> is called a Cartesian-closed category.

The following categories are Cartesian closed: the category of sets, the category of small categories 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====
<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>