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Stream: learning: questions

Topic: algebra morphisms parameterised by endofunctor morphisms

Nathanael Arkor (Apr 25 2023 at 10:12):

Suppose $s$ and $t$ are endo-1-cells in a 2-category, $(h, \lambda)$ is a lax morphism of endo-1-cells, and $(c, \gamma)$ and $(d, \delta)$ are algebras for $s$ and $t$ respectively. There's a natural notion of morphism of algebras, parameterised by $(h, \lambda)$: namely, we ask for a pair $(f, \phi)$ satisfying the compatibility law pictured.
image.png

This seems like it either ought to be a known concept, or be an example of a more general one. Does anyone know of a name or reference for such morphisms?

Amar Hadzihasanovic (Apr 25 2023 at 11:31):

If you see an algebra in this sense as classified by a functor $A \to \mathbf{B}$, where $A$ is free on

• two 0-cells $0, 1$,
• two 1-cells $a: 0 \to 1$ and $e: 1 \to 1$, and
• a 2-cell $\alpha: e \circ a \Rightarrow a$,

then the data $(f, \phi, h, \lambda)$ is precisely a 1-cell from $(s, c, \gamma)$ to $(t, d, \delta)$ in the 2-category $\mathbf{B} \Leftarrow A$, where $- \Leftarrow A$ is right adjoint to the lax Gray right tensoring $- \otimes A$.
In this sense it is one of the 2-categorically natural notions of morphism, the other ones being those arising from right adjoints to the left tensoring $A \otimes -$ and the symmetric “pseudo” Gray tensoring $A \otimes_\mathrm{ps} -$.

Mike Shulman (Apr 25 2023 at 14:42):

"Right adjoint to the lax Gray right tensoring" is a rather highfalutin' way to say "The 2-category of 2-functors, lax natural transformations, and modifications". (-:

Amar Hadzihasanovic (Apr 25 2023 at 15:37):

Yeah, but I can never remember which ones are lax and which ones are oplax natural transformations, whereas I know how Gray products work :D

Nathanael Arkor (Apr 25 2023 at 15:47):

I agree the definition may be motivated more abstractly, though I'm particularly interested in references that study such morphisms in the context of algebras for endo-1-cells or monads.

Mike Shulman (Apr 25 2023 at 20:24):

Surely the "lax" Gray tensor is left adjoint to the category of lax transformations, and similarly for the oplax ones?

Amar Hadzihasanovic (Apr 25 2023 at 21:24):

Afaik people only qualify the non-symmetric Gray product with "lax" to distinguish it from the "pseudo Gray tensor product" which is symmetric; the category of lax transformations is right adjoint to tensoring with the "lax" product on the right, and the one of oplax transformations to tensoring on the left.
I guess one could achieve the same effect by defining the "oplax Gray tensor product" as $A \otimes^\mathrm{op} B := B \otimes A$ and having lax transformations adjoint to $- \otimes A$ and oplax transformations to $- \otimes^\mathrm{op} A$, but I've never seen anyone do it.