Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: Approach to whiskering with lax functors

Tobias Schmude (Apr 26 2023 at 10:34):

In this n-category café post Mike explained that it's not possible to whisker 2-transformations with lax functors:
Given 2-categories, 2-functors and a 2-transformation as in the following diagram
whiskering_setup.png
we can construct the prewhiskering $\alpha*E$ by just restricting $\alpha$ along $E$:
prewhiskering_incorrect.png
For the postwhiskering $H*\alpha$ we encounter a problem though. The diagram
postwhiskering_incorrect.png
does not compose to a component 2-morphism $(H*\alpha)_b$ filling the square since the compositors point inwards. Using oplax functors/transformations clearly does not solve the problem either.

This is strange, since in the 1-dimensional case we can view natural transformations as functors, and whiskering just as composition of functors:
Given the same setup as above, just with 1-categories, 1-functors and a 1-transformation, we can view $\alpha$ equivalently as a functor $\overline{\alpha}: \mathcal{B} \times \mathrm{arr} \to \mathcal{C}$, where $\mathrm{arr}$ is the walking arrow $S \to T$. The source and target functors of $\alpha$ are then given by $F = \overline{\alpha}(-, S), G = \overline{\alpha}(-, T)$ and the components of the transformation are given by $\alpha_B = \overline{\alpha}(B, \to)$. Functoriality of $\overline{\alpha}$ then comes down to functoriality of $F$ and $G$ as well as naturality of $\alpha$. The whiskerings are then given by $\overline{\alpha*E} = \overline{\alpha} \circ (E \times \mathrm{arr})$ and $\overline{H*\alpha} = H \circ \overline{\alpha}$.

Observation: the functor $\overline{\alpha}$ not only supplies the components $\alpha_B$. For each morphism $b$, the diagonal of the corresponding naturality square is part of the data as $\overline{\alpha}(b, \to)$.

We can already see how this categorifies: in a suitable definition of lax 2-transformation, we're going to have the diagonal as part of the data along with 2-morphisms filling the triangles. To be fully explicit, we have the components
components_correct.png
The prewhiskering is given by restriction as usual, and the postwhiskering now works because in the diagram
postwhiskering_correct.png
the 2-morphisms pointing inwards towards the diagonal compose.
Of course there are coherence axioms to take care of, but this doesn't lead to problems since lax functors compose, just as in the 1-dimensional case.

Tobias Schmude (Apr 26 2023 at 10:35):

Concerning this I have two questions:

• I'd guess that this approach is too natural to be new. Any idea as to if/where people have used this already?
• I'd like to have some reality checks for this definition. Are there applications of lax 2-transformations where I could check if this notion makes sense?

Amar Hadzihasanovic (Apr 26 2023 at 11:01):

How do you propose to, for example, compose these “natural transformations” vertically?
That is, if $\alpha: F \Rightarrow G$ and $\beta: G \Rightarrow H$ are natural transformations in your sense, how would you define $\beta \circ \alpha: F \Rightarrow H$?

Tobias Schmude (Apr 26 2023 at 12:07):

Good point! Extending the definition, we could have arbitrary zigzags of 2-morphisms filling the naturality square. I don't see how this fits the transformations-as-functors pov though.