User:Fropuff/Draft 4

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1 Hom functors
- 1.1 Dictionary
2 Comma categories
- 2.1 Slice category
  - 2.1.1 Limits and colimits
  - 2.1.2 Examples
- 2.2 Coslice category
  - 2.2.1 Limits and colimits
  - 2.2.2 Examples

[edit] Hom functors

[edit] Dictionary

Let $f:A_1\to A_2$ and $g:B_1\to B_2$ be morphisms between objects in a category C. Let A and B be objects in C. Then we can define the following objects:

$Hom(A, B)$	set
$Hom(f, B)$	function	$\mathrm{Hom}(A_2,B)\to\mathrm{Hom}(A_1,B)\,$	$\phi\mapsto \phi\circ f$
$Hom(A, g)$	function	$\mathrm{Hom}(A,B_1)\to\mathrm{Hom}(A,B_2)\,$	$\phi\mapsto g\circ\phi$
$Hom(f, g)$	function	$\mathrm{Hom}(A_2,B_1)\to\mathrm{Hom}(A_1,B_2)\,$	$\phi\mapsto g\circ\phi\circ f$
$Hom(A, - )$	functor	$\mathcal C\to \mathbf{Set}$	$B\mapsto\mathrm{Hom}(A,B)$ $g\mapsto\mathrm{Hom}(A,g)$
$Hom( - , B)$	functor	$\mathcal C^{\mathrm{op}}\to \mathbf{Set}$	$A\mapsto\mathrm{Hom}(A,B)$ $f\mapsto\mathrm{Hom}(f,B)$
$Hom(f, - )$	natural transformation	$\mathrm{Hom}(A_2,-)\to\mathrm{Hom}(A_1,-)\,$	$\mathrm{Hom}(f,B)\,$
$Hom( - , g)$	natural transformation	$\mathrm{Hom}(-,B_1)\to\mathrm{Hom}(-,B_2)\,$	$\mathrm{Hom}(A,g)\,$
$Hom( - , - )$	bifunctor	$\mathcal C^{\mathrm{op}}\times\mathcal C\to \mathbf{Set}$	$(A,B)\mapsto\mathrm{Hom}(A,B)$ $(f,g)\mapsto\mathrm{Hom}(f,g)$
$Hom( - 1, - 2)$	functor	$\mathcal C^{\mathrm{op}}\to \mathbf{Set}^{\mathcal C}$	$A\mapsto\mathrm{Hom}(A,-)$ $f\mapsto\mathrm{Hom}(f,-)$
$Hom( - 2, - 1)$	functor	$\mathcal C\to \mathbf{Set}^{\mathcal C^{\mathrm{op}}}$	$B\mapsto\mathrm{Hom}(-,B)$ $g\mapsto\mathrm{Hom}(-,g)$

[edit] Comma categories

comma category (T ↓ S)

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

[edit] Slice category

Let C be a category and let A be an object in C. The slice category is denoted (C ↓ A) or C/A.

objects are morphisms to A in C, e.g. f : X → A
morphisms are commutative triangles φ : (f : X → A) → (g : Y → A) with f = g∘φ

The forgetful functor, U : C/A → C, assigns to each morphism f : X → 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 × Y → A). U then commutes with colimits.

[edit] Limits and colimits

If I is an initial object in C then (I → A) is an initial object in C/A.
The coproduct of f_X and f_Y is the natural morphism f_X+Y.

(id_A : A → A) is a terminal object in C/A.
Products in C/A are pullbacks in C.

[edit] 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/ℕ is the category of graded sets (morphisms must preserve the grade, so perhaps different than a multiset)

[edit] Coslice category

Let C be a category and let A be an object in C. The coslice category is denoted (A ↓ C) or A/C.

objects are morphisms from A in C, e.g. f : A → X
morphisms are commutative triangles φ : (f : A → X) → (g : A → Y) with g = φ∘f.

[edit] Limits and colimits

[edit] Examples

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

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User:Fropuff/Draft 4

From Wikipedia, the free encyclopedia

Contents

[edit] Hom functors

[edit] Dictionary

[edit] Comma categories

[edit] Slice category

[edit] Limits and colimits

[edit] Examples

[edit] Coslice category

[edit] Limits and colimits

[edit] Examples

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