nLab
comma category

Context

Category theory

Limits and colimits

Contents

Idea

The comma category of two functors f:CEf : C \to E and g:DEg : D \to E is like an arrow category of EE where all arrows have their source in the image of ff and their target in the image of gg (and the morphisms between arrows keep track of how these sources and targets are in these images). It is a kind of 2-pullback: a directed refinement of the homotopy pullback of two functors between groupoids.

Definition

We discuss three equivalent definitions of comma categories

  1. Explicitly in components

  2. As a fiber product

  3. As a 2-pullback

Remark

The terminology “comma category” is a holdover from the original notation (f,g)(f,g) for such a category, which generalises (x,y)(x,y) or C(x,y)C(x,y) for a hom-set.

Another common notation for the comma category is (fg)(f\downarrow g). The original notation, from which the terminology is derived, is (f,g)(f,g), but this is rarely used any more.

In components

Definition

If f:CEf:C\to E and g:DEg:D\to E are functors, their comma category is the category (f/g)(f/g) whose

  • objects are triples (c,d,α)(c,d,\alpha) where cCc\in C, dDd\in D, and α:f(c)g(d)\alpha:f(c)\to g(d) is a morphism in EE, and whose

  • morphisms from (c 1,d 1,α 1)(c_1,d_1,\alpha_1) to (c 2,d 2,α 2)(c_2,d_2,\alpha_2) are pairs (β,γ)(\beta,\gamma), where β:c 1c 2\beta:c_1\to c_2 and γ:d 1d 2\gamma:d_1\to d_2 are morphisms in CC and DD, respectively, such that α 2.f(β)=g(γ).α 1\alpha_2 . f(\beta) = g(\gamma) . \alpha_1.

f(c 1) f(β) f(c 2) α 1 α 2 g(d 1) g(γ) g(d 2) (c 1,d 1,α 1) (β,γ) (c 2,d 2,α 2) \array{ f(c_1) &\stackrel{f(\beta)}{\to}& f(c_2) \\ \downarrow^{\alpha_1} && \downarrow^{\alpha_2} \\ g(d_1) &\stackrel{g(\gamma)}{\to}& g(d_2) \\ \\ (c_1,d_1, \alpha_1) &\stackrel{(\beta,\gamma)}{\to}& (c_2,d_2, \alpha_2) }
  • composition of morphisms is given on components by composition in CC and DD.

As a fiber product

Let I={ab}I = \{a \to b\} be the (directed) interval category and E I=Funct(I,E)E^I = Funct(I,E) the functor category.

The comma category is the pullback

(f/g) E I d 0×d 1 C×D f×g E×E \array{ (f/g) &\to& E^I \\ \downarrow && \downarrow^{\mathrlap{d_0 \times d_1}} \\ C \times D &\stackrel{f \times g}{\to}& E \times E }

(in the 1-category Cat of categories).

Compare this with the construction at generalized universal bundle and with the definition of loop space object.

As a 2-pullback

Alternatively, the comma category is the “lax pullback” – or rather the comma object (see the discussion at 2-limit) of the pullback diagram,

C f D g E \array{ && C \\ && \downarrow^f \\ D &\stackrel{g}{\to}& E }

i.e. the universal cone that commutes up to a natural transformation

(f/g) C f D g E \array{ (f/g) &\to& C \\ \downarrow &\swArrow& \downarrow^f \\ D &\stackrel{g}{\to}& E }

In terms of the imagery of loop spaces objects, the comma category is the category of directed paths in EE which start in the image of ff and end in the image of gg.

Examples

  • If ff and gg are both the identity functor of a category CC, then (f/g)(f/g) is the category C 2C ^{\mathbf{2}} of arrows in CC.

  • If ff is the identity functor of CC and gg is the inclusion 1C1\to C of an object cCc\in C, then (f/g)(f/g) is the slice category C/cC/c.

  • Likewise if gg is the identity and ff is the inclusion of cc, then (f/g)(f/g) is the coslice category c/Cc/C.

Properties

2-categorical properties

The comma category (f/g)(f/g) comes with a canonical 2-cell in the square

Comma Square ( f / g ) (f/g) C C D D E E f f g g α \alpha

which is universal in the 2-category Cat; that is, it is an example of a 2-limit (in fact, it is a strict 2-limit). Squares with the same universal property in an arbitrary 2-category are called comma squares and their top left vertex is called a comma object.

Functors and comma categories

See at

Further reading

a low-tech description with several special cases identified in somewhat archaic terminology

Revised on August 15, 2013 03:39:01 by Anonymous Coward (82.226.206.220)