nLab comma category

Idea

The comma category of two functors $f : C \to E$ and $g : D \to E$ is a category like an arrow category of $E$ where all arrows have their source in the image of $f$ and their target in the image of $g$ (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

• Explicitly in components

• As a pullback

• As a 2-limit

Remark

The terminology “comma category” is a holdover from the original notation $(f,g)$ for such a category, which generalises $(x,y)$ or $C(x,y)$ for a hom-set. This is rarely used any more. More common modern notations for the comma category are $(f/g)$, which we will use on this page, and $(f\downarrow g)$.

Via components: the objectwise definition

Definition

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

• objects are triples $(c,d,\alpha)$ where $c\in C$, $d\in D$, and

  (1)$$\alpha \colon f(c) \to g(d)$$

  is a morphism in $E$,

and whose

• morphisms from $(c_1,d_1,\alpha_1)$ to $(c_2,d_2,\alpha_2)$ are pairs $(\beta,\gamma)$, where $\beta:c_1\to c_2$ and $\gamma:d_1\to d_2$ are morphisms in $C$ and $D$, respectively, such that $\alpha_2 . f(\beta) = g(\gamma) . \alpha_1$.

$$\array{ f(c_1) &\stackrel{f(\beta)}{\rightarrow}& f(c_2) \\ \downarrow^{\alpha_1} && \downarrow^{\alpha_2} \\ g(d_1) &\stackrel{g(\gamma)}{\to}& g(d_2) \\ \\ (c_1,d_1, \alpha_1) &\stackrel{(\beta,\gamma)}{\to}& (c_2,d_2, \alpha_2) }$$

• composition of morphisms is given on components by composition in $C$ and $D$.

Remark

In addition, there are two canonical forgetful functors defined on the comma category:

• there is a functor $H_C\colon (f/g)\rightarrow C$ which sends each object $(c,d,\alpha)$ to $c$, and each pair $(\beta,\gamma)$ to $\beta$.

• there is a functor $H_D\colon (f/g)\rightarrow D$ which sends each object $(c,d,\alpha)$ to $d$, and each pair $(\beta,\gamma)$ to $\gamma$.

Furthermore:

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

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

Remark

Alternatively, requiring the morphisms (1) to be isomorphisms yields the notion of an iso-comma category (an iso-comma object in Cat). This has the universal property of a non-lax “2-pullback”.

As a pullback in the 1-category of categories

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

The comma category is the pullback

$$\array{ (f/g) &\to& E^I \\ \big\downarrow & \scriptsize{(pb)} & \big\downarrow \mathrlap{^{\mathrlap{(F\mapsto F(a))\times(F\mapsto F(b))}}} \\ C \times D & \underset{f \times g}{\longrightarrow} & E \times E }$$

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.

As a 2-limit in the 2-category of categories

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

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

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):

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

(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 $H_C$ and $H_D$ 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 $E$ which start in the image of $f$ and end in the image of $g$.

Examples

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

• If $f$ is the identity functor of $C$ and $g$ is the inclusion $1\to C$ of an object $c\in C$, then $(f/g)$ is the slice category $C/c$.

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

• A natural transformation $\tau \colon F \to G$ with $F,G :\colon C\to D$ may be regarded as a functor $T \colon C\to (F/G)$ with $T(c)=(c,c,\tau_c)$ and $T(f)=(f,f)$. Conversely, any such functor $T$ such that the two projections from $(F/G)$ back to $C$ are both left inverses for $T$ yields a corresponding natural transformation. This is an expression of the universal property of $(F/G)$ as a comma object.

Properties

Completeness and cocompleteness

If $C$ and $D$ are cocomplete and $f: C \to E$ is cocontinuous and $g: D \to E$ is an arbitrary functor (not necessarily cocontinuous) then the comma category $(f/g)$ is cocomplete. Similarly, as $(f/g)^{op}\cong (g^{op}/f^{op})$, if $C$ and $D$ are complete and $g: D \to E$ is continuous and $f: C \to E$ is an arbitrary functor (not necessarily continuous) then the comma category $(f/g)$ is complete.

For a proof, see

• Rydeheard, David E., and Rod M. Burstall. Computational category theory. Vol. 152. Englewood Cliffs: Prentice Hall, 1988. Section 5.2: colimits in comma categories.

Functors and comma categories

• functors and comma categories

Related concepts

• Quillen's theorem A
• comma object

References

The notion of comma categories was introduced (in order to characterize adjoint functors, but without giving them a name) in:

• F. W. Lawvere, Functorial Semantics of Algebraic Theories, Ph.D. thesis, Columbia University (1963)

• Consolato Pellegrino, Creazione di limiti nelle categorie comma, Rivista di Matematica della Università di Parma 3 (1974): 201-204.

• Consolato Pellegrino, Un teorema di completezza per le categorie comma, sue applicazioni agli n-grafi, Atti del Seminario Matematico e Fisico dell’Università di Modena 23 (1974): 223-230.

Textbook accounts:

• Saunders MacLane, §II.6 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 1.6 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]

See also:

• Toby Bartels, Comma categories