nLab colax/lax comma object

Idea

For a 2-monad $T$, the 2-category $T Alg$ of $T$-algebras and pseudo $T$-morphisms has weak 2-limits (bilimits), and more precisely pie-limits. The 2-categories $T Alg_l$ and $T Alg_c$ of lax and colax $T$-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 $T$-algebra structure. Of these one of the most commonly encountered is a comma object $(f/g)$ where $f$ is colax and $g$ 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 $T$ be a (strict, for simplicity) 2-monad on a (strict, for simplicity) 2-category $K$, and let $f:A\to C$ be a colax $T$-morphism and $g:B\to C$ a lax $T$-morphism. Suppose that the (strict, for simplicity) comma object $(f/g)$ exists in $K$; thus it is equipped with projections $p:(f/g)\to A$ and $q:(f/g)\to B$ (which are, so far, only morphisms in $K$) and a 2-cell

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

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

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

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

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

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

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

Examples

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