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Stream: theory: category theory

Topic: Monads induced by a section/retraction pair

Patrick Nicodemus (Jan 08 2023 at 19:59):

I have the following conjecture. Let $\mathcal{C}$ be a nice category, say a topos. Let $s: A\to B$, $r : B\to A$ such that $rs=\operatorname{id}$. There are pullback functors $r^\ast : \mathcal{C}/A\to\mathcal{C}/B$, $s^\ast : \mathcal{C}/B\to\mathcal{C}/A$. $r^\ast$ has a left and right adjoint - left adjoint is just postcomposition with $r$, right adjoint is more complicated and relies on $\mathcal{C}$ being locally Cartesian closed. Let us call the left and right adjoint $\amalg_r, \Pi_r$ respectively.

My conjecture is that $\amalg_r\circ\Pi_s$ is a monad on $\mathcal{C}/A$. This morning I worked out the unit and I am thinking about the multiplication right now. In the case of Sets this essentially just boils down to the fact that every object is a monoid for the coproduct and so $X+(-)$ is a monad for any $X$. I have a specific example of this that I care about in SSet, there I have checked the details but the argument doesn't seem to really rely on anything too specific to SSet.

My question is, is this known, where is it written down, who should I cite, etc. If it is correct, does it more generally hold when the codomain fibration is replaced with a sufficiently nice Grothendieck fibration where left and right adjoints to reindexing functors exist and satisfy a beck-chevalley condition

Jean-Baptiste Vienney (Jan 08 2023 at 21:36):

I don't understand most of what you wrote but that sounds however a little bit familiar to me. If you consider in the category of modules on a $\mathbb{Q}$-algebra (eg. a field of characteristic $0$) the tensor algebra $T(A)$ and the symmetric algebra $S(A)$ of vector spaces $A$, then you have a section/retractation pair
$(r;s):T(A) \rightarrow S(A) \rightarrow T(A)$ and $S$ and $T$ are monads. What does it give if you apply your conjecture? (what are $\amalg_r\circ\Pi_s$ and $\mathcal{C}/A$?)

Jean-Baptiste Vienney (Jan 08 2023 at 21:38):

$r$ takes $x_{1} \otimes ... \otimes x_{n}$ to $x_{1} \otimes_{s} ... \otimes_{s} x_{n}$ and $s$ takes $x_{1} \otimes_{s} ... \otimes_{s} x_{n}$ to $\frac{1}{n!} \underset{\sigma \in \mathfrak{S}_{n}}{\sum}x_{\sigma(1)} \otimes ... \otimes x_{\sigma(n)}$

Jean-Baptiste Vienney (Jan 08 2023 at 21:45):

Well, maybe I'll try myself to understand what you wrote and see what it gives!

Patrick Nicodemus (Jan 08 2023 at 21:56):

I actually am no longer convinced this works, I am going to try and strengthen my assumptions and consider the case where we have a pointed object $b_0: 1\to B$ and the section-retraction pair is of the form $s(a) = (a,b_0)$, $r(a,b)=a$, exhibiting $A$ as a retract of $A\times B$.

Patrick Nicodemus (Jan 08 2023 at 21:57):

The specific case in which I am interested in is in the category of simplicial sets, where we take $A$ to be some simplicial set and consider the inclusion $A\to A\times I$ at time $t=1$. Here I have done the computations and checked that there is indeed a monad structure

Fernando Yamauti (Jan 08 2023 at 22:00):

Patrick Nicodemus said:

I have the following conjecture. Let $\mathcal{C}$ be a nice category, say a topos. Let $s: A\to B$, $r : B\to A$ such that $rs=\operatorname{id}$. There are pullback functors $r^\ast : \mathcal{C}/A\to\mathcal{C}/B$, $s^\ast : \mathcal{C}/B\to\mathcal{C}/A$. $r^\ast$ has a left and right adjoint - left adjoint is just postcomposition with $r$, right adjoint is more complicated and relies on $\mathcal{C}$ being locally Cartesian closed. Let us call the left and right adjoint $\amalg_r, \Pi_r$ respectively.

My conjecture is that $\amalg_r\circ\Pi_s$ is a monad on $\mathcal{C}/A$. This morning I worked out the unit and I am thinking about the multiplication right now. In the case of Sets this essentially just boils down to the fact that every object is a monoid for the coproduct and so $X+(-)$ is a monad for any $X$. I have a specific example of this that I care about in SSet, there I have checked the details but the argument doesn't seem to really rely on anything too specific to SSet.

My question is, is this known, where is it written down, who should I cite, etc. If it is correct, does it more generally hold when the codomain fibration is replaced with a sufficiently nice Grothendieck fibration where left and right adjoints to reindexing functors exist and satisfy a beck-chevalley condition

I'm not sure if being a retract is relevant to your case since you have not revealed the specific example you have in mind. Still, it looks like you are talking about something like a polynomial functor.

Patrick Nicodemus (Jan 08 2023 at 22:11):

Thank you. Not to be overly obscure, as I said I am interested in the case of the inclusion of a simplicial set $A$ into the cylinder $A\times I$ at time $t=1$ and the inverse given by projection onto $A$. There is a monad on $\mathbf{SSet}/A$ arising from these maps

Patrick Nicodemus (Jan 08 2023 at 22:40):

I don't know, forget it. I wrote down this monad and it seems too nice and simple to depend on too many subtleties of simplicial sets, but I can't come up with a more abstract way to construct it.