# nLab hom-object

### Context

#### Enriched category theory

enriched category theory

mapping space

# Contents

## Idea

An ordinary locally small category $C$ has for any ordered pair of objects $x,y$ a hom-set $C(x,y)$—an object in the category $Set$.

For $C$ more generally an enriched category over a closed monoidal category $V$, there is – by definition – for all $x,y$ an object $C(x,y) \in obj V$ that plays the role of the “collection of morphisms” from $x$ to $y$

## Examples

• The category Grpd of groupoids is a enriched over itself. Hence for any two groupoids $A,B$, there is a hom-groupoid $Grpd(A,B)$. This is the functor category $Func(A,B)$.
homotopycohomologyhomology
$[S^n,-]$$[-,A]$$(-) \otimes A$
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space $\mathbb{R}Hom(S^n,-)$cocycles $\mathbb{R}Hom(-,A)$derived tensor product $(-) \otimes^{\mathbb{L}} A$

Revised on June 29, 2012 02:04:28 by Urs Schreiber (89.204.139.141)