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.
We discuss three equivalent definitions of comma categories
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 $(f\downarrow g)$.
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
is a morphism in $E$,
and whose
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:
These functors and natural transformation together give the comma category a 2-category theoretic universal property; see this section for more.
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”.
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
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.
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
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):
(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 $g$.
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.
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
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:
Last revised on November 13, 2024 at 17:33:06. See the history of this page for a list of all contributions to it.