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Stream: theory: category theory

Topic: strictifying maps with isofibrancy

Matteo Capucci (he/him) (Apr 11 2024 at 10:03):

This question is related to #theory: category theory > Cat(Mon(Cat)) vs Mon(Cat(Cat)), where @Nathanael Arkor pointed out that a monoidal double category is a double monoidal category whose source and target maps are strict monoidal.
I was wondering whether requiring that the double monoidal category is isofibrant might be enough to recover an equivalent monoidal double category.
The idea is to replace the monoidal double category of loose arrows $A_1$ with a different one obtained by using the isofibrancy of $\langle s,t \rangle:A_1 \to A_0\times A_0$.

In general, this seems to suggest that if $A,B$ are $T$-algebras, for $T$ 2-monad on $\bf Cat$, and $f:A \to B$ is an isofibration which is also a pseudomap of algebras, we can strictify it by replacing $\alpha:TA \to A$ with a different $\alpha'$ for which $f$ is then a strict map. The new algebra structure $\alpha'$ is obtained by pulling back $\alpha$ along the isomorphism:
image.png

Matteo Capucci (he/him) (Apr 11 2024 at 10:04):

Is this is a known technique? Can I conclude $\alpha'$ does indeed satisfy the same axioms as $\alpha$?

El Mehdi Cherradi (Apr 11 2024 at 13:51):

In the settings of tribes (here $\mathbf{Cat}$ with fibrations the isofibrations), this can be seen as an instance of the "straightening lemma" in Joyal's notes, but this is not connected to the matter of algebras. I would however expect the functor
$\mathbf{Ho}(\mathcal{T} \text{-Alg}) \to \mathbf{Ho}(\mathcal{T})\text{-Alg}$
(from the homotopy category of algebras on an $\infty$-monad $\mathcal{T}$ to the category of algebras of the derived monad) to be weakly smothering.

Matteo Capucci (he/him) (Apr 11 2024 at 14:15):

Thanks for your interest! But I need you to unpack this bit. What would weakly smothering imply?

El Mehdi Cherradi (Apr 11 2024 at 14:50):

Sorry, here I would say the insight is that the functor is full (and essentially surjective). Combined with a suitable description of the homotopy category, this says that the morphism given by $f$ in $\mathbf{Ho}(\mathcal{T})\text{-Alg}$ lifts to a (zigzag of) strict ones (here this could be a span where the left leg is a surjective equivalence and the right one an isofibration). This does not tell you exactly how to obtain a new algebra structure on $A$ though, but rather "factors" $f$ through a algebra on $A^\simeq$ (the category of isomorphisms in $A$). I hope this is more precise but I did not mean it as a formal statement with proof.

Mike Shulman (Apr 11 2024 at 15:23):

I don't recall having seen exactly this before, but it sounds plausible. You might have to require $T$ to be a flexible 2-monad, though (roughly, one that doesn't impose equalities between operations on objects) -- it probably wouldn't work for strict monoidal structures.

Mike Shulman (Apr 11 2024 at 15:24):

You might even have to require $T$ to satisfy the strictification theorem, e.g. your modified $\alpha$ might a priori only be a pseudoalgebra structure.

Matteo Capucci (he/him) (Apr 11 2024 at 15:31):

Indeed, I was expecting some restriction should be placed on $T$. Thanks for weighing in, Mike!