User:Fropuff/Drafts/Comma category

comma category (T &darr; S)


 * $$\begin{matrix} T(\alpha) & \xrightarrow{T(g)} & T(\alpha')\\ f \Bigg\downarrow & & \Bigg\downarrow f'\\ S(\beta) & \xrightarrow[S(h)]{} & S(\beta') \end{matrix}$$


 * hom-set category (A &darr; B) = Hom(A, B) as a discrete category
 * morphism (or arrow) category (C &darr; C) = C2
 * (U &darr; A), objects U over A, or morphisms from U to A
 * slice category, objects over A, written (C &darr; A) or C/A
 * (&Delta; &darr; F) category of cones to F
 * (A &darr; U), objects U under A, or morphisms from A to U
 * coslice category, objects under A, written (A &darr; C) or A/C
 * (F &darr; &Delta;) category of cones from F

Slice category
Let C be a category and let A be an object in C. The slice category is denoted (C &darr; A) or C/A.
 * objects are morphisms to A in C, e.g. f : X &rarr; A
 * morphisms are commutative triangles &phi; : (f : X &rarr; A) &rarr; (g : Y &rarr; A) with f = g&#x2218;&phi;

The forgetful functor, U : C/A &rarr; C, assigns to each morphism f : X &rarr; A its domain X. If C has finite products this functor has a right-adjoint which assigns to each space Y the projection map (A &times; Y &rarr; A). U then commutes with colimits.

Limits and colimits

 * If I is an initial object in C then (I &rarr; A) is an initial object in C/A.
 * The coproduct of fX and fY is the natural morphism fX+Y.


 * (idA : A &rarr; A) is a terminal object in C/A.
 * Products in C/A are pullbacks in C.

Examples

 * If A is terminal, then C/A is isomorphic to C.
 * If C is a poset category, C/A is the principal ideal of objects less than A.
 * Set/&#x2115; is the category of graded sets (morphisms must preserve the grade, so perhaps different than a multiset)

Coslice category
Let C be a category and let A be an object in C. The coslice category is denoted (A &darr; C) or A/C.
 * objects are morphisms from A in C, e.g. f : A &rarr; X
 * morphisms are commutative triangles &phi; : (f : A &rarr; X) &rarr; (g : A &rarr; Y) with g = &phi;&#x2218;f.

Examples

 * If A is initial, then A/C is isomorphic to C.
 * &bull;/Set is the category of pointed sets
 * &bull;/Top is the category of pointed spaces
 * R/CRing is the category of commutative R-algebras