Contravariant hom functor
Concept history
A revision trail for this concept page.
Revision 591
7/25/2026, 10:09:29 AM · Ancient Tree
Concept edited
Compare with revision 5892 changed lines
1
Let $\mathcal{C}$ be a [[Locally small category|locally small category]]. For each object $B$ in $\mathcal{C}$, there is a [[Functor|functor]]2
$$\operatorname{Hom}_{\mathcal{C}}(-,B):\mathcal{C}^{\operatorname{op}}\rightarrow \mathbf{Set}$$3
defined by:4
- On objects: each object $X$ is sent to the set $\operatorname{Hom}_{\mathcal{C}}(X,B)$.5
- On morphisms: a morphism $f:X\rightarrow Y$ is sent to the function6
$$f^*=\operatorname{Hom}_{\mathcal{C}}(f, B): \operatorname{Hom}_{\mathcal{C}}(Y, B) \rightarrow \operatorname{Hom}_{\mathcal{C}}(X, B)$$7
given by pre-composition: $g\mapsto g\circ f$.7
given by [[Pre-composition|pre-composition]]: $g\mapsto g\circ f$.Revision 589
7/25/2026, 10:08:10 AM · Ancient Tree
Concept created
Let $\mathcal{C}$ be a [[Locally small category|locally small category]]. For each object $B$ in $\mathcal{C}$, there is a [[Functor|functor]]
$$\operatorname{Hom}_{\mathcal{C}}(-,B):\mathcal{C}^{\operatorname{op}}\rightarrow \mathbf{Set}$$
defined by:
- On objects: each object $X$ is sent to the set $\operatorname{Hom}_{\mathcal{C}}(X,B)$.
- On morphisms: a morphism $f:X\rightarrow Y$ is sent to the function
$$f^*=\operatorname{Hom}_{\mathcal{C}}(f, B): \operatorname{Hom}_{\mathcal{C}}(Y, B) \rightarrow \operatorname{Hom}_{\mathcal{C}}(X, B)$$
given by pre-composition: $g\mapsto g\circ f$.