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: learning: questions

Topic: Factorization system over a subcategory

Cole Comfort (Nov 24 2020 at 16:31):

In her paper Distributive laws for Lawvere theories, Cheng describes factorizations systems between categories $L$ and $R$ up to zig-zags in some shared wide subcategory $J$. When $J$ contains different kinds of isomorphisms, then depending on which isomorphisms are chosen, this is known to correspond to different types of distributive laws.

However, she goes on to show that when $L,R$ are Lawvere theories, then if one picks out $J:={\sf FinOrd}^{\sf op}$, then a generalized factorization system in this sense is the same as a distributive law on $L \otimes_{{\sf FinOrd}^{\sf op}} R$ in some category of internal profunctors. Does anyone know if this generalizes to arbitrary $J$, or is this just a property of the subcategory ${\sf FinOrd}^{\sf op}$ of Lawvere theories, and of groupoids? I am particularly interested in the case when $J$ is a category of spans of monomorphisms.

John Baez (Nov 24 2020 at 16:45):

I don't understand this stuff, especially "in some category of internal profunctors". But can you say the specific conjecture you'd be shooting for? Something like a distributive law between the prop $\mathsf{FinInj}$ where morphisms are monomorphisms (between finite ordinals) and the prop $\mathsf{FinInj}^{\mathrm{op}}$?

Cole Comfort (Nov 24 2020 at 16:57):

I have just been naively assuming that there is always some sort of correspondence between these sort of generalized factorization systems and so called "relaxed" distributive laws. And I am wondering what fails, if indeed something fails when one changes what $J$ is.

On second thought, what I said I wanted is not actually what I want in this case.

I am interested in a very particular thing, but I will try to see if I can distill it to a level which seems well-motivated.

John Baez (Nov 24 2020 at 17:01):

So my specific question is not what you're talking about? That's what I got from you mentioning "spans of monos".

John Baez (Nov 24 2020 at 17:02):

I think the distributive law I alluded to, between the prop $\mathsf{FinInj}$ where morphisms are monomorphisms (between finite ordinals) and the prop $\mathsf{FinInj}^{\mathrm{op}}$, should exist.

Cole Comfort (Nov 24 2020 at 17:25):

I think that in the case you alluded to, you just need $J$ to be all isomorphisms, and then the distributive law is witnessed by the bone equation.

For a much more contrived example, I am looking at a similar situation except where there are extra subobjects lying around that aren't properly objects.

Fix some natural $n>1$. Consider the subcategory ${\sf sub}_n$ of ${\sf Span}^\sim(\sf FinOrd)$ generated by spans of the form $n^k \xleftarrow{m} \ell \xrightarrow{m} n^k$ for all $m$ monic and for all natural numbers $k,\ell$.

And consider another subcategory of isomorphisms between powers of $n$, ${\sf iso}_n$.

There is a distributive law ${\sf sub}_n \otimes {\sf iso}_n$, yielding a category ${\sf subiso}_n$, seen as adding the subobjects to the isos.

Consider another subcategory ${\sf mon}_n$ generated by spans of the form $n^k = n^k \xrightarrow{m} n^\ell$ for $m$ monic.

Then there is another distributive law ${\sf subiso}_n \otimes_{{\sf iso}_n} {\sf mon}_n$ yielding a category ${\sf submon}_n$, seen as adding the subobjects to the monos.

So there is some generalized factorization system with $L={\sf submon}_n^{\sf op}$, $R={\sf submon}_n$ and $J= {\sf subiso}_n$, and I am not sure if this actually corresponds to a distributive law on ${\sf submon}_n^{\sf op} \otimes_{{\sf subiso}_n } {\sf submon}_n$, but I highly suspect that it does!

I should note that all of this is going on inside ${\sf Mon}$-$\sf Prof$.