nLab
colax/lax comma object

Colax/lax comma object

Idea

For a 2-monad TT, the 2-category TAlgT Alg of TT-algebras and pseudo TT-morphisms has weak 2-limits (bilimits), and more precisely pie-limits. The 2-categories TAlg lT Alg_l and TAlg cT Alg_c of lax and colax TT-morphisms do not have all 2-limits (even weak ones), but they do have some, particularly when some of the morphisms involved in the diagram are strict or pseudo; see rigged limit for a characterization of these. There is no 2-category containing both lax and colax morphisms, but nevertheless some limits of diagrams involving lax and colax morphisms can be given a TT-algebra structure. Of these one of the most commonly encountered is a comma object (f/g)(f/g) where ff is colax and gg is lax. It is unclear exactly how to state a universal property for this comma object, but it is probably related to the double category of algebras.

Definition

Let TT be a (strict, for simplicity) 2-monad on a (strict, for simplicity) 2-category KK, and let f:ACf:A\to C be a colax TT-morphism and g:BCg:B\to C a lax TT-morphism. Suppose that the (strict, for simplicity) comma object (f/g)(f/g) exists in KK; thus it is equipped with projections p:(f/g)Ap:(f/g)\to A and q:(f/g)Bq:(f/g)\to B (which are, so far, only morphisms in KK) and a 2-cell

(f/g) p A q α f B g C\array{ (f/g) & \xrightarrow{p} & A\\ ^q\downarrow & \swArrow_\alpha & \downarrow^f \\ B & \xrightarrow{g} & C }

that is universal among such 2-cells. Now consider the following pasting composite:

T(f/g) TA Tα Tf TB TC f¯ A g¯ f B g C\array{ T(f/g) & \to & T A \\ \downarrow & \swArrow_{T\alpha} & \downarrow^{T f} & \searrow \\ T B & \to & T C & \swArrow_{\bar{f}} & A \\ & \searrow & \swArrow_{\bar{g}} & \searrow &\downarrow^f \\ && B & \xrightarrow{g} & C }

Here f¯\bar{f} is the colax TT-morphism constraint of ff, while g¯\bar{g} is the lax TT-morphism constraint of gg. Notice that these go in exactly the right directions for the above pasting to be well-defined. Now by the universal property of (f/g)(f/g), there is a unique morphism T(f/g)(f/g)T(f/g) \to (f/g) such that the above pasting composite is equal to the following one:

T(f/g) TA TB (f/g) A α f B g C \array{ T(f/g) & \to & T A \\ \downarrow & \searrow & & \searrow \\ T B & & (f/g) & \to & A \\ & \searrow & \downarrow & \swArrow_\alpha &\downarrow^f \\ && B & \xrightarrow{g} & C }

where the empty quadrilaterals commute. A similar argument shows that this map T(f/g)(f/g)T(f/g) \to (f/g) is the action map of a TT-algebra structure on (f/g)(f/g), such that the projections p:(f/g)Ap:(f/g)\to A and q:(f/g)Bq:(f/g)\to B are strict TT-morphisms and α\alpha is a 2-cell in the double category of algebras TAlgT \mathbf{Alg}. (In fact, the latter assertion is precisely the equality of the above two pasting diagrams.)

The strictness of the projections p,qp,q is familiar from the behavior of rigged limits in TAlg lT Alg_l and TAlg cT Alg_c. However, it is unclear exactly what universal property this TT-algebra (f/g)(f/g) has, although it seems likely to involve the double category TAlgT \mathbf{Alg} somehow.

Examples

  • Some comma double categories are colax/lax comma objects in the double category of double categories (and lax and colax double functors).

Created on February 27, 2018 at 11:43:52. See the history of this page for a list of all contributions to it.