For a 2-monad , there are several naturally defined 2-categories of -algebras, depending on whether we take the morphisms to be lax, colax (oplax), strong/pseudo, or strict. Of these, the lax and colax cases and are the most general, since a strict morphism is also strong, and a strong morphism can be regarded as either lax or colax (using the structure morphisms or their inverses).
There is no natural 2-category that contains both lax and colax morphisms, since a lax morphism cannot be composed with a colax morphism and obtain any sort of -morphism. However, lax and colax -morphisms can be combined as the horizontal and vertical morphisms in a double category. This double category allows a characterization of strong morphisms as companion pairs, of doctrinal adjunctions as conjoint pairs, and its 2-cells are relevant for defining oplax/lax comma objects.
Let be a 2-monad on a 2-category . The objects of the double category are the -algebras, the horizontal morphisms are the lax -morphisms, and the vertical morphisms are the colax -morphisms. The 2-cells are 2-cells in between composites of underlying morphisms, such that a certain cube of structure 2-cells commutes: consider a square
where horizontal arrows are lax -morphism, and vertical ones are colax. A 2-cell fills the square and it is such that the equality of pasting diagrams
holds. This means that a certain diagram of 2-cells, that can be obtained translating the above equality into a commutative hexagon, is commutative.
For a concrete example in the case of monoidal categories, take lax monoidal functors and and colax monoidal functors and , this cube becomes two commutative hexagons for the composition and unit constraints:
If is a strict 2-category, then is a strict double category; otherwise it is “pseudo in both directions” in some sense (such double categories can be defined, but are tricky; see double category). As the objects we can use either strict -algebras (if is a strict 2-monad) or pseudo algebras (if is either a strict or a pseudo 2-monad).
The horizontal 2-category of is the 2-category of lax -morphisms, and its vertical 2-category is .
There is a forgetful double functor from to the double category of quintets in . In fact, can be enhanced to a triple category whose “transversal morphisms” are strict -morphisms (or pseudo ones, if is only pseudo, in which case we have to deal with “triply pseudo triple categories”), and the induced forgetful functor from this triple category to the triple category of “quintets and commutative cubes” in is monadic in an appropriate sense.
A companion pair in consists of two isomorphic morphisms in between a pair of -algebras along with a pseudo -morphism structure on one (hence both). The category of companion pairs between two -algebras is thus equivalent (though not isomorphic) to the category of pseudo -morphisms.
A conjoint pair in is precisely a “colax/lax” or doctrinal adjunction, and the theorems about doctrinal adjunction can naturally be expressed as lifting properties of the forgetful functor from to .
The universal 2-cell of an oplax/lax comma object has the structure of a square in . Its universal property ought to be related to as well, but this is still unclear.
Last revised on August 7, 2024 at 10:01:00. See the history of this page for a list of all contributions to it.