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 a category 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 can also be seen a kind of 2-limit: a directed refinement of the homotopy pullback of two functors between groupoids.

Definition

We discuss three equivalent definitions of comma categories

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. This is rarely used any more. More common modern notations for the comma category are (f/g)(f/g), which we will use on this page, and (fg)(f\downarrow g).

Via components: the objectwise definition

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.

The definition of (f/g)(f/g) is now complete. In addition, there are two canonical forgetful functors defined on the comma category:

  • there is a functor H C:(f/g)CH_C\colon (f/g)\rightarrow C which sends each object (c,d,α)(c,d,\alpha) to cc, and each pair (β,γ)(\beta,\gamma) to β\beta.

  • there is a functor H D:(f/g)DH_D\colon (f/g)\rightarrow D which sends each object (c,d,α)(c,d,\alpha) to dd, and each pair (β,γ)(\beta,\gamma) to γ\gamma.

Furthermore:

  • there is a natural transformation θ:fH CgH D\theta : f \circ H_C \to g\circ H_D defined by θ (c,d,α)=α\theta_{(c,d,\alpha)} = \alpha.

These functors and natural transformation together give the comma category a 2-categorical universal property; see this section for more.

Via fiber products in the 1-category Cat

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 (pb) (FF(a))×(FF(b)) C×D f×g E×E \array{ (f/g) &\to& E^I \\ \downarrow & (pb) & \downarrow^{\mathrlap{(F\mapsto F(a))\times(F\mapsto F(b))}} \\ C \times D &\stackrel{f \times g}{\to}& E \times E }

in the standard sense of pullback of morphisms in the 1-category Cat of categories.

Compare this with the construction of homotopy pullback (here), hence with the definition of loop space object and also with generalized universal bundle.

Via 2-category theory: as a 2-limit

The comma category is the comma object of the cospan CfEgDC\overset{f}{\rightarrow}E\overset{g}{\leftarrow}D in the 2-category CatCat. This means it is an appropriate weighted 2-categorical limit (in fact, a strict 2-limit) of the diagram

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

Specifically, it is the universal span making the following square commute up to a specified natural transformation (such a universal square is in general called a comma square):

(f/g) H C C H D f D g E \array{ (f/g) &\overset{H_C}{\to}& C \\ \mathllap{{}^{H_D}} \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ D &\stackrel{g}{\to}& E }

(Sometimes this is called a “lax pullback”, but that terminology properly refers to something else; see comma object and 2-limit.)

Notably, the forgetful functors H CH_C and H DH_D from the “objectwise” definition are thus recovered via a categorical construction: they are the projections from the summit of the “appropriate” 2-categorical limit.

In terms of the imagery of loop space 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

Functors and comma categories

References

Revised on July 6, 2017 12:44:42 by Mike Shulman (76.167.222.204)