comma double category




A comma double category is a generalization of a comma category to double categories and virtual double categories.

There are two apparently-maximal levels of generality of this construction, which intersect nontrivially but do not coincide. The first is an ordinary comma object in the 2-category of virtual double categories (which includes as a full sub-2-category the 2-category of pseudo double categories and lax functors), which produces another virtual double category. The second is a colax/lax comma object relative to the 2-monad whose algebras are pseudo double categories. The two constructions intersect in the case of the comma object of a pseudo double functor over a lax one, between pseudo double categories.


For virtual double categories

For virtual double categories, the comma virtual double category has the universal property of a comma object in the 2-category of virtual double categories, functors and vertical transformations.

Explicitly, it is constructed as follows. Let C,D,EC, D, E be virtual double categories and F:CE,G:DEF : C \to E, G : D \to E be functors of virtual double categories. The comma double category F/GF / G is defined as

  1. Its vertical category is the ordinary comma category F v/G vF_v/G_v of the vertical components of the functors. We write an object as A:FA CGA DA : F A_C \to G A_D.
  2. A horizontal arrow RR from AA to BB consists of horizontal arrows R C:A CB CR_C : A_C \to B_C and R D:A DB DR_D : A_D \to B_D and a 2-cell in EE from FR CF R_C to GR DG R_D along A,BA,B.
  3. A 2-cell α\alpha from R 1,,R nR_1,\ldots,R_n to SS consists of a pair of 2-cells α C:(R 1C,,R nC)S C\alpha_C : (R_{1C},\ldots,R_{nC}) \to S_C and α D:(R 1D,,R nD)S D\alpha_D : (R_{1D},\ldots,R_{nD}) \to S_D such that G(α D)(R 1,,R n)=SF(α C)G(\alpha_D) \circ (R_1,\ldots,R_n) = S \circ F(\alpha_C).

The oplax/lax case

Let K=Cat K = Cat^{\rightrightarrows} be the 2-category of directed graphs internal to Cat. There is a 2-monad TT on KK whose algebras are (pseudo) double categories, and whose lax and colax morphisms are lax and colax double functors respectively. The oplax/lax comma double category is then an oplax/lax comma object for this 2-monad.

Structures on a virtual comma double category

Next, we consider what properties are required of the input data (in the virtual case) to determine that a comma virtual double category has units and composites. An analogy with the double category case gives some guidance. Since functors of virtual double categories correspond to lax functors of double categories, we don’t have any requirements for the functor GG on top of DD having composites or units. On the other hand, for FF to be “oplax”, we require that it be normal for units or furthermore strong for composites.


If CC, DD and EE have units and FF is a normal functor, then F/GF / G has units.


If CC, DD and EE have composites and FF is a strong functor, then F/GF / G has composites.

Next, the comma has restrictions whenever the constituent categories do and the functors preserve them.


If CC, DD and EE have restrictions and F,GF,G preserve them, then F/GF / G has restrictions.

In practice this proof burden can be reduced if we are interested in virtual equipments (i.e. having units and restrictions) because in that case restrictions are automatically preserved. We summarize this as follows:


If C,DC, D and EE are virtual equipments and FF is a normal functor, then F/GF/G is a virtual equipment.


  1. A monad TT in the horizontal bicategory of a double category CC is equivalent to a lax functor T:1CT : 1 \to C from the terminal double category. In this case we might call Id C/TId_C / T the slice double category?.

  2. generalized multicategories can be constructed using a slice category when the monad TT is a polynomial monad. Specifically, let CC be the double category of polynomial functors in some locally cartesian closed category EE; then a polynomial monad TT on EE can be identified with a horizontal monad in CC on the terminal object 11. The slice Id C/TId_C/T is then equivalent to the “horizontal Kleisli category” presented in Cruttwell-Shulman; TT-multicategories are then monads in that comma double category.

  3. The double category of decorated cospans? is naturally constructed as a comma double category. Given a symmetric lax monoidal functor F:(C,+)(D,)F : (C,+) \to (D,\otimes), there is an associated lax double functor from F:Cospan(C)BDF' : Cospan(C) \to BD where BDBD is the delooping of DD into a double category whose horizontal category is DD and vertical category is the terminal category. Then there is a colax (pseudo even) double functor *:1BD* : 1 \to BD that picks out the unique object of BDBD. Then the double category of decorated cospans is */F*/F'.

  4. poset-valued sets given by an endofunctor FF on RelRel and a poset PP can be viewed as the comma double category from FF to PP, since a poset is a monad in RelRel, and FF is a colax endofunctor of RelRel. The “morphisms” of poset-valued sets are the horizontal morphisms in the resulting comma double category.

  5. The Dialectica construction? associated to an internal poset Ω\Omega in a monoidal category CC with pullbacks can be obtained as a comma double category. Let Span(C)Span(C) be the double category whose horizontal morphisms are spans in CC, regard C×C opC\times C^{op} as a double category in the horizontal direction, and let F:C×C opSpan(C)F: C\times C^{op} \to Span(C) be the colax functor defined on objects by (A,B)AB(A,B) \mapsto A\otimes B and taking the pair (f,g):(A 1,B 1)(A 2,B 2)(f,g) : (A_1,B_1) \to (A_2,B_2) (so that f:A 1A 1f:A_1\to A_1 and g:B 2B 1g:B_2\to B_1) to the span A 1B 11gA 1B 2f1A 2B 2A_1\otimes B_1 \xleftarrow{1\otimes g} A_1\otimes B_2 \xrightarrow{f\otimes 1} A_2\otimes B_2. The internal poset Ω\Omega is a monad in Span(C)Span(C), so we have a comma double category F/ΩF/\Omega, whose horizontal category is the Dialectica construction Dial(C,Ω)Dial(C,\Omega).


An explicit description of the comma double category from an oplax to a lax functor is given in

  • Kenny Courser?, A bicategory of decorated cospans (arxiv)

Last revised on April 6, 2018 at 16:33:16. See the history of this page for a list of all contributions to it.