nLab
hom-set

Contents

Definition

For locally small categories

Given objects xx and yy in a locally small category, the hom-set hom(x,y)hom(x,y) is the collection of all morphisms from xx to yy. In a closed category, the hom-set may also be called the external hom to distinguish it from the internal hom.

For enriched categories

For a category CC enriched over a category VV, the “hom-set” C(x,y)C(x,y) is an object of VV, the hom-object.

For internal categories

For C=(C 0,C 1,s,t,e,c)C = (C_0, C_1, s,t,e, c) an internal category, the generalized objects of CC are morphisms x:XC 0x: X \to C_0 and y:YC 0y: Y \to C_0, and the “hom-set” becomes the pullback C(x,y)C(x,y) in

C(x,y) Y y X C 1 t C 0 x s C 0 \array{ C(x,y) & \to & Y \\ \downarrow & \searrow & & \searrow^{y} \\ X & & C_1 & \stackrel{t}\to & C_0 \\ & \searrow^{x} & \downarrow_s \\ & & C_0 }

In particular, in a category with a terminal generator **, we may identitfy morphisms x,y:*C 0x,y: * \to C_0 with global objects of CC and form C(x,y)C(x,y) as above.

Revised on November 28, 2014 08:59:35 by Urs Schreiber (82.224.164.72)