For a 2-monad , the 2-category of -algebras and pseudo -morphisms has weak 2-limits (bilimits), and more precisely pie-limits. The 2-categories and of lax and colax -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 -algebra structure. Of these one of the most commonly encountered is a comma object where is colax and 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.
Let be a (strict, for simplicity) 2-monad on a (strict, for simplicity) 2-category , and let be a colax -morphism and a lax -morphism. Suppose that the (strict, for simplicity) comma object exists in ; thus it is equipped with projections and (which are, so far, only morphisms in ) and a 2-cell
that is universal among such 2-cells. Now consider the following pasting composite:
Here is the colax -morphism constraint of , while is the lax -morphism constraint of . Notice that these go in exactly the right directions for the above pasting to be well-defined. Now by the universal property of , there is a unique morphism such that the above pasting composite is equal to the following one:
where the empty quadrilaterals commute. A similar argument shows that this map is the action map of a -algebra structure on , such that the projections and are strict -morphisms and is a 2-cell in the double category of algebras . (In fact, the latter assertion is precisely the equality of the above two pasting diagrams.)
The strictness of the projections is familiar from the behavior of rigged limits in and . However, it is unclear exactly what universal property this -algebra has, although it seems likely to involve the double category somehow.
Last revised on November 1, 2024 at 10:05:34. See the history of this page for a list of all contributions to it.