User:Fropuff/Drafts/Hom functor

Official page: Hom functor

Properties

Preservation of limits

Covariant Hom functors preserve all limits. In particular, they preserve all small limits, and are therefore continuous. By duality, the contravariant Hom functors take colimits to limits. Covariant Hom functors do not necessarily preserve colimits.

Given a diagram F : J → C and an object X of C the limit of composite functor Hom(X, F–) : J → Set is given by the set of all cones from X to F:

lim Hom(X, F–) = Cone(X, F)

The limiting cone is given by the maps

$\pi _{i}:\mathrm {Cone} (X,F)\to \mathrm {Hom} (X,F_{i})$

where $\pi _{i}(\psi )=\psi _{i}$. If F has a limit in C then Hom(X, lim F) is naturally isomorphic to the set of all cones from X to F so that

Hom(X, lim F) = lim Hom(X, F–)

Moreover, the Hom functor Hom(X, –) takes the limiting cone of F to the limiting cone of Hom(X, F–). It follows that Hom(X, –) preserves the limits of F.

Categorical objects associated with Hom sets

The are great variety of objects associated with Hom sets. These are summarized in the following table. In this table

• C is a category,
• A, A₁, A₂, and B, B₁, B₂ are objects in C,
• $f:A_{1}\to A_{2}$ and $g:B_{1}\to B_{2}$ are morphisms in C.

Object	Type	Domain and range	Definition
$\mathrm {Hom} (A,B)\,$	set
$\mathrm {Hom} (f,B)\,$	function	$\mathrm {Hom} (A_{2},B)\to \mathrm {Hom} (A_{1},B)\,$	$\phi \mapsto \phi \circ f$
$\mathrm {Hom} (A,g)\,$	function	$\mathrm {Hom} (A,B_{1})\to \mathrm {Hom} (A,B_{2})\,$	$\phi \mapsto g\circ \phi$
$\mathrm {Hom} (f,g)\,$	function	$\mathrm {Hom} (A_{2},B_{1})\to \mathrm {Hom} (A_{1},B_{2})\,$	$\phi \mapsto g\circ \phi \circ f$
$\mathrm {Hom} (A,-)\,$	functor	${\mathcal {C}}\to \mathbf {Set}$	$B\mapsto \mathrm {Hom} (A,B)$ $g\mapsto \mathrm {Hom} (A,g)$
$\mathrm {Hom} (-,B)\,$	functor	${\mathcal {C}}^{\mathrm {op} }\to \mathbf {Set}$	$A\mapsto \mathrm {Hom} (A,B)$ $f\mapsto \mathrm {Hom} (f,B)$
$\mathrm {Hom} (f,-)\,$	natural transformation	$\mathrm {Hom} (A_{2},-)\to \mathrm {Hom} (A_{1},-)\,$	$\mathrm {Hom} (f,B)\,$
$\mathrm {Hom} (-,g)\,$	natural transformation	$\mathrm {Hom} (-,B_{1})\to \mathrm {Hom} (-,B_{2})\,$	$\mathrm {Hom} (A,g)\,$
$\mathrm {Hom} (-,-)\,$	bifunctor	${\mathcal {C}}^{\mathrm {op} }\times {\mathcal {C}}\to \mathbf {Set}$	$(A,B)\mapsto \mathrm {Hom} (A,B)$ $(f,g)\mapsto \mathrm {Hom} (f,g)$
$\mathrm {Hom} (-_{1},-_{2})\,$	functor	${\mathcal {C}}^{\mathrm {op} }\to \mathbf {Set} ^{\mathcal {C}}$	$A\mapsto \mathrm {Hom} (A,-)$ $f\mapsto \mathrm {Hom} (f,-)$
$\mathrm {Hom} (-_{2},-_{1})\,$	functor	${\mathcal {C}}\to \mathbf {Set} ^{{\mathcal {C}}^{\mathrm {op} }}$	$B\mapsto \mathrm {Hom} (-,B)$ $g\mapsto \mathrm {Hom} (-,g)$