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Stream: theory: category theory

Topic: Enriched double categories

Patrick Nicodemus (Feb 10 2024 at 16:45):

Let $K$ be a 2-category equipped with a 2-monad $T$. Then we can talk about lax, strong, colax, ... morphisms of $T$ algebras. My question is:

In the setting of $V$-enriched category theory, if $T$ is a $V$-enriched monad, can we give a general setting to talk about a different kind of morphisms between $T$ algebras?

My initial idea was something like the following formalism: Take the underlying $1$-category of $K$, say
$|K|$. We can form a kind of "enriched double category" structure on $K$, where $|K|$ is taken as the "object of objects" in the usual sense of double category theory, and to each pair of morphisms in $K$ we associate a $V$-object of "arrows" between them. When $V$ is Cat and $K$ is a 2-category, this could be the category of lax 2-cells between the two morphisms. The interchange law for vertical composition and horizontal composition would perhaps require a symmetric monoidal category.

Patrick Nicodemus (Feb 10 2024 at 16:51):

Maybe it not need be an "enriched double category", maybe it can be an ordinary double category, and then the question is "given a V-enriched category what are some techniques for constructing double category structures on it".

Patrick Nicodemus (Feb 10 2024 at 18:52):

For example I would want to have a setting where, if V is the category of bicategories, then we could talk about a morphism of algebras which is a commutative square up to an adjoint equivalence.