For a 2-monad $T$, there are several naturally defined 2-categories of $T$-algebras, depending on whether we take the morphisms to be lax, colax (oplax), strong/pseudo, or strict. Of these, the lax and colax cases $T Alg_l$ and $T Alg_c$ 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 $T$-morphism. However, lax and colax $T$-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 $T$ be a 2-monad on a 2-category $\mathcal{K}$. The objects of the double category $T \mathbf{Alg}$ are the $T$-algebras, the horizontal morphisms are the lax $T$-morphisms, and the vertical morphisms are the colax $T$-morphisms. The 2-cells are 2-cells in $\mathcal{K}$ between composites of underlying morphisms, such that a certain cube of structure 2-cells commutes: consider a square
where horizontal arrows are lax $T$-morphism, and vertical ones are colax. A 2-cell $\alpha : g f \Rightarrow k h$ 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 $F:A\to B$ and $K:C\to D$ and colax monoidal functors $G:B\to D$ and $H:A\to C$, this cube becomes two commutative hexagons for the composition and unit constraints:
If $\mathcal{K}$ is a strict 2-category, then $T \mathbf{Alg}$ is a strict double category; otherwise it is â€śpseudo in both directionsâ€ť in some sense (such double categories can be defined, but are tricky; see double category). As the objects we can use either strict $T$-algebras (if $T$ is a strict 2-monad) or pseudo algebras (if $T$ is either a strict or a pseudo 2-monad).
The horizontal 2-category of $T \mathbf{Alg}$ is the 2-category $T Alg_l$ of lax $T$-morphisms, and its vertical 2-category is $T Alg_c$.
There is a forgetful double functor from $T \mathbf{Alg}$ to the double category $\mathbf{Q}(\mathcal{K})$ of quintets in $\mathcal{K}$. In fact, $T \mathbf{Alg}$ can be enhanced to a triple category whose â€śtransversal morphismsâ€ť are strict $T$-morphisms (or pseudo ones, if $T$ is only pseudo, in which case we have to deal with â€śtriply pseudo triple categoriesâ€ť), and the induced forgetful functor from this triple category to the triple category of â€śquintets and commutative cubesâ€ť in $\mathcal{K}$ is monadic in an appropriate sense.
A companion pair in $T \mathbf{Alg}$ consists of two isomorphic morphisms in $\mathcal{K}$ between a pair of $T$-algebras along with a pseudo $T$-morphism structure on one (hence both). The category of companion pairs between two $T$-algebras is thus equivalent (though not isomorphic) to the category of pseudo $T$-morphisms.
A conjoint pair in $T \mathbf{Alg}$ is precisely a â€ścolax/laxâ€ť or doctrinal adjunction, and the theorems about doctrinal adjunction can naturally be expressed as lifting properties of the forgetful functor from $T \mathbf{Alg}$ to $\mathbf{Q}(\mathcal{K})$.
The universal 2-cell of an oplax/lax comma object has the structure of a square in $T \mathbf{Alg}$. Its universal property ought to be related to $T \mathbf{Alg}$ 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.