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Stream: theory: category theory

Topic: A new monad from a strong monad

Patrick Nicodemus (Feb 01 2023 at 05:12):

I have come across something in my research (this has to do with natural weak factorization systems introduced by Grandis and Tholen) and I would like to know if this construction is known.

Let $C$ be a category with finite limits. $1$ is the terminal object. Let $T$ be a monad. We assume $T$ is strong with respect to the Cartesian product, so there is a map $\alpha_{X,Y}: T(X)\times Y\to T(X\times Y)$.

Fix $Y$ an object in $C$. I claim that $T$ induces a monad $R$ on the slice category $C/Y$. I will sketch some components of $R$.

If $f : X\to Y$ is an object in the slice category, then $\operatorname{dom}(R(f))$ is the pullback of $\alpha_{1,Y}$ and $T(f)$, and the body of $R(f)$ is the composition of $\alpha_{1,Y}^\ast(T(f))$ with $\pi_Y$.

To construct the unit $\operatorname{dom}(R(f))$, by the universal property of the pullback it suffices to give maps $X\to Y, X\to T(1), X\to T(X)$ satisfying certain coherence conditions. For the first we take $f$, for the second map we take $\eta_1 \circ !_X$, for the third map we take $\eta_X$, where $\eta$ is the unit of the monad $T$.

For the multiplication it is more work and I will procrastinate writing it down here unless someone asks for it.

Is this monad discussed in the literature? Where can I read more about it?

Morgan Rogers (he/him) (Feb 01 2023 at 10:01):

I'm by no means the expert, but this is the first time I've seen someone pulling back along the strength of a strong monad in a construction. For the unit, the coherence condition is that $\alpha_{1,Y} \circ \langle \eta_1 \circ !_X,f \rangle = T(f) \circ \eta_X$; is that necessarily satisfied?

Patrick Nicodemus (Feb 01 2023 at 15:04):

Morgan Rogers (he/him) said:

I'm by no means the expert, but this is the first time I've seen someone pulling back along the strength of a strong monad in a construction. For the unit, the coherence condition is that $\alpha_{1,Y} \circ \langle \eta_1 \circ !_X,f \rangle = T(f) \circ \eta_X$; is that necessarily satisfied?

Yes, here is a proof.

Patrick Nicodemus (Feb 01 2023 at 15:06):

image.png
I have introduced the abbreviation $E(f) = \operatorname{dom}(R(f))$

Bryce Clarke (Feb 02 2023 at 08:47):

This construction reminds me of something similar. Take a monad $T$ on a category $C$ with pullbacks. Then you can construction a monad on the arrow category $C^\mathbf{2}$ by sending a morphism $f \colon A \rightarrow B$ to the morphism obtained by pulling back $T(f) \colon TA \rightarrow TB$ along the unit $\eta_{B} \colon B \rightarrow TB$.

Bryce Clarke (Feb 02 2023 at 08:47):

If your strong monad happens to preserve the terminal object, this construction gives you the same monad.

Bryce Clarke (Feb 02 2023 at 08:49):

What are algebras for the monad you construct? Do you have a particular strong monad in mind?

Patrick Nicodemus (Feb 03 2023 at 00:00):

I found a hole in my argument when I wrote it down as I cannot prove one of the coherence conditions. I am generalizing from a very concrete case and trying to find some reasonable degree of generality in which this construction holds.

I believe the following assumption will help me complete the proof: if $f:X\to Z$ then the naturality square for the strength $\alpha$ applied to $(f, \operatorname{id} _Y)$ is a pullback.

Is there anywhere this condition is studied?

Patrick Nicodemus (Feb 03 2023 at 00:04):

@Bryce Clarke Yes. I am working with the cone monad on simplicial sets, the day convolution with the singleton space. An algebra for the cone monad is a contractible simplicial set and my goal is to adapt this to the slice category over an arbitrary simplicial set Y in order to give a notion of a space which is contractible in the slice category, basically fiber homotopy equivalent to the identity map id_Y in a very strong sense.

Patrick Nicodemus (Feb 03 2023 at 00:37):

to clarify, this construction definitely works for simplicial sets +the cone monad. just trying to generalize a bit.

Patrick Nicodemus (Feb 04 2023 at 01:36):

I spent more time working on this problem today and I believe I was off on the wrong foot previously.

Patrick Nicodemus (Feb 04 2023 at 01:36):

Let me start over.

Patrick Nicodemus (Feb 04 2023 at 01:39):

Let $C$ be a category with finite limits. . Let $T$ be a monad. We assume $T$ is strong with respect to the Cartesian product, so there is a map $\alpha_{X,Y}: T(X)\times Y\to T(X\times Y)$. For $f : X\to Y$, let $\operatorname{gr}(f)$ be the graph of $f$, the map $(1_X, f) : X\to X\times Y$.

Fix $Y$ an object in $C$. I conjecture that $T$ induces a monad $R$ on the slice category $C/Y$. I will sketch some components of $R$.

If $f : X\to Y$ is an object in the slice category, then $E(f)=\operatorname{dom}(R(f))$ is the pullback of $\alpha_{X,Y}$ and $T(\operatorname{gr}(f))$, and the body of $R(f)$ is the composition of $\alpha_{X,Y}^\ast(T(\operatorname{gr}(f)))$ with $\pi_Y$.

To construct the unit $\operatorname{dom}(R(f))$, by the universal property of the pullback it suffices to give two maps $X\to TX$ and one map $X\to Y$ satisfying certain coherence conditions. For the maps $X\to TX$ we take $\eta$, for the map $X\to Y$ we take $f$.

Patrick Nicodemus (Feb 04 2023 at 01:41):

Lemma, I claim that the following square commutes.

Patrick Nicodemus (Feb 04 2023 at 01:41):

image.png

Patrick Nicodemus (Feb 04 2023 at 01:43):

Proof: let $!_Y$ denote the unique map from $Y$ to the terminal object, and apply naturality of $\alpha$ to the maps $1_X, !_Y$. Then use the rule for strong monads that "strengthening with the unit does nothing."

Patrick Nicodemus (Feb 04 2023 at 01:48):

Oh forget it. Here's a pdf I latexed.

Patrick Nicodemus (Feb 04 2023 at 01:50):

misc1.pdf

Patrick Nicodemus (Feb 04 2023 at 01:50):

I have not yet checked the unit and associativity laws for the monad, just constructed the unit and multiplication as natural transformations.

Patrick Nicodemus (Feb 04 2023 at 01:50):

So this could still fail

Patrick Nicodemus (Feb 04 2023 at 01:54):

but we will be optimistic. this time it is all going to work out

Patrick Nicodemus (Feb 04 2023 at 01:54):

image.png

Patrick Nicodemus (Feb 04 2023 at 01:56):

This is the same thing I was trying to figure out a month ago here. https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/Monads.20induced.20by.20a.20section.2Fretraction.20pair/near/320113811
So I will be happy to get this resolved.

Patrick Nicodemus (Feb 04 2023 at 20:19):

Yes! I have now written out the proofs that the multiplication and unit laws hold.

Patrick Nicodemus (Feb 04 2023 at 20:32):

There are many situations in homological algebra where one can use monads to identify a distinguished class of homologically or homotopically trivial objects, such as flasque sheaves, injective sheaves, projective or injective objects, free modules and so on. This is very useful. However model category theory as a formalism seems to care more about distinguished maps than distinguished objects, so it seems important to be able to translate a concept of "contractible object" from a monad on a category to a broader notion of "acyclic fibration" in terms of a monad on the slice category.

Patrick Nicodemus (Feb 04 2023 at 20:35):

In my case, I know that contractible simplicial objects are algebras of the cone monad, so I have been trying to adapt the cone monad so that it gives a monad on the slice category over some fixed simplicial set, which would allow us to identify objects "contractible in the slice category" = homotopy equivalent to id_Y.

Patrick Nicodemus (Feb 05 2023 at 06:37):

I am very irritated right now. My monad preserves pullbacks and I did not know there was a well developed theory of monads that preserve pullbacks. I have ignored papers about "Cartesian monads" because a Cartesian monad, I presumed, is one that preserves Cartesian products or perhaps all finite limits because a Cartesian category is one with all finite limits. How was I supposed to know a Cartesian monad is one that preserves pullbacks? That is a silly naming convention.

Patrick Nicodemus (Feb 05 2023 at 06:50):

Oh wait. It seems the terminology is inconsistent. Johnstone uses the term in the way I expected him to, for him a Cartesian monad is indeed one which preserves all finite limits. But the nlab page says it means a monad preserving pullbacks. I will have to go back through these papers on Cartesian monads I ignored earlier and figure out which way they are using the term.

Patrick Nicodemus (Feb 05 2023 at 10:30):

Sorry for all the questions.
Is there anything known about Cartesian monads (or parametric right adjoint monads) and strengths? Are there conditions that guarantee that a Cartesian monad has a strength for the Cartesian product or conversely?

Morgan Rogers (he/him) (Feb 05 2023 at 14:07):

Questions are what this place is all about, don't apologise! The reason for the difference in convention is one of slight laziness: many of the nice properties of monads which preserve finite limits only actually depend on the preservation of pullbacks.
I think you're likely to find something if you consider adjunctions rather than monads (if you pick a monadic adjunction you won't lose anything) since the "Frobenius reciprocity condition" relates something that looks a lot like a strength to a structure preservation condition for the functors in the adjunction. Sorry for being vague, I'm on a train and only have a vague picture of this in my mind, but check the section on cartesian closedness in Part A of the Elephant.

James Deikun (Feb 07 2023 at 13:15):

Patrick Nicodemus said:

Sorry for all the questions.
Is there anything known about Cartesian monads (or parametric right adjoint monads) and strengths? Are there conditions that guarantee that a Cartesian monad has a strength for the Cartesian product or conversely?

The one I know about is that every [[polynomial monad]] has a canonical strength and is strongly Cartesian.

Additionally, I notice that the cone monad has very similar characteristics to the simplicial set monad on semisimplicial sets--they are both (strongly?) Cartesian and both have a right adjoint.

Patrick Nicodemus (Feb 07 2023 at 23:21):

Thank you. After some thought I have narrowed down my question a bit further. The cone monad is a polynomial functor whose "signature" is of the form

1 <- 1 -> I -> 1
where I is the unit interval and the arrow 1->I is the inclusion at time t=1.
Note that because right adjoints preserve limits and in particular terminal objects, the middle leg of this functor must send 1 to I. Thus T(1)=I. Making this identification, the resulting map 1-> T(1) is the unit of the monad.

Now fix some object Y and take the cartesian product of the original signature with Y.

Y <- Y-> YxT(1)-> Y

This induces a polynomial endofunctor on SSet/Y which can be seen as a kind of lifting of the cone monad to the slice category over Y. As it happens this is also a monad. So my question is, why? under what conditions does the original polynomial monad structure lift to the slice category in this fashion

Patrick Nicodemus (Feb 07 2023 at 23:25):

So more precisely, fix a locally cartesian closed category C with finite limits and consider a monad T such that T is precisely the polynomial monad arising from the signature
$1\xrightarrow{\eta_1}T(1)\to 1$
Under what additional assumptions does the induced polynomial functor

$Y\to Y\times T(1)\to Y$ inherit a monad structure for all $Y$?

Patrick Nicodemus (Feb 07 2023 at 23:28):

In this case it happens that $\Delta_{\eta_1}\circ\Pi_{\eta_1 }$ is the identity. This assumption is apparently necessary to construct the unit on induced monad.

Patrick Nicodemus (Feb 13 2023 at 07:25):

I have some partial results in this question but I cannot get the comonad of the natural weak factorization, just the monad. I am about ready to give up on this question and move onto something else.
But it is interesting at least that the monad exists in a variety of circumstances.

Patrick Nicodemus (Feb 13 2023 at 07:26):

I wish I could obtain the comomad. I have a natural weak factorization system on simplicial sets and i am absolutely stuck trying to generalize it to any other setting. I will post what I have soon

Patrick Nicodemus (Feb 13 2023 at 19:50):

Hi folks. I am posting a latexed version of my recent work here. I would very much like anybody's help on answering this question I have.

misc2.pdf

Patrick Nicodemus (Feb 13 2023 at 19:51):

Here is the short version.

Patrick Nicodemus (Feb 13 2023 at 19:58):

Let $\mathcal{C}$ be a category with finite limits. Let $(T,\eta, \mu)$ be a monad on $\mathcal{C}$ which is strong with respect to the Cartesian product by a natural isomorphism $\alpha_{X,Y}: T(X)\times Y\to T(X\times Y)$.

Patrick Nicodemus (Feb 13 2023 at 19:59):

Then there is an induced monad $R$ on the slice category $\mathcal{C}/Y$ (for arbitrary $Y$). The object part of $R$ is described in the image below:

Patrick Nicodemus (Feb 13 2023 at 19:59):

image.png

Patrick Nicodemus (Feb 13 2023 at 20:00):

Here, $\operatorname{gr}(f)$ is the graph of $f$, i.e., $(1,f) : X\to X\times Y$.

Patrick Nicodemus (Feb 13 2023 at 20:00):

The unit and multiplication are described in the link misc2.pdf.

Patrick Nicodemus (Feb 13 2023 at 20:02):

Now, again, let $\mathcal{C}$ be a category with finite limits, but this time assume that $\mathcal{C}$ is locally Cartesian closed. Let $I$ be an object of $\mathcal{C}$ and $e : 1\to I$ a distinguished morphism, and let $T$ be defined as the polynomial functor arising from the diagram

$1 \leftarrow 1 \xrightarrow{e} I\to 1$

Patrick Nicodemus (Feb 13 2023 at 20:04):

Assume now that $T$ is endowed with the structure of a polynomial monad, in particular with Cartesian unit and multiplication. (Actually for free we get the Cartesian unit by the definition, it's only the multiplication that has to be assumed.)

Patrick Nicodemus (Feb 13 2023 at 20:05):

In this case, for any object $Y$, the slice category $\mathcal{C}/Y$ can be endowed with a polynomial monad $R$, defined by the polynomial diagram

$Y = Y \xrightarrow{1,e} Y\times I\xrightarrow{\pi_Y}Y$

Patrick Nicodemus (Feb 13 2023 at 20:06):

Well, calling $R$ a monad is a lie as I have only constructed the multiplication and unit so far, not checked the unit and associativity. suffice it to say I strongly suspect they hold. The reason I have not checked them is I have a bigger problem I am trying to solve.

Patrick Nicodemus (Feb 13 2023 at 20:07):

Both of these constructions, when we take $\mathcal{C}$ to be simplicial sets and $T$ to be the cone monad, reduce to the same monad $R$ on $\mathbf{SSet}/Y$, which is a kind of fibered version of the cone monad.

Patrick Nicodemus (Feb 13 2023 at 20:08):

What is interesting is that in this case of simplicial sets, you have not just a monad on the slice category but a comonad on the coslice category, i.e. you have a natural weak factorization system in the sense of Grandis and Tholen.

Patrick Nicodemus (Feb 13 2023 at 20:10):

So I have tried to give two generalizations of this simplicial set construction in slightly different directions. In both directions I was able to construct the monad of the natural weak factorization system but I cannot figure out how to construct the comultiplication of the comonad.

Patrick Nicodemus (Feb 13 2023 at 20:13):

What I am asking, of anyone who cares to listen, is : What are the extra categorical conditions that hold for $SSet$ that allow us to construct the comultiplication of the comonad? What extra categorical assumptions do we have to throw in to get a n. w. f. s?

Patrick Nicodemus (Feb 13 2023 at 20:14):

I absolutely do not believe that this construction relies on any unique distinguishing properties of the category of simplicial sets, other than being a very nice category. But I am sure we should be able to get something similar in any elementary topos or probably even an LCCC with sufficient assumptions on the monad.

James Deikun (Feb 19 2023 at 13:04):

Just for intuition's sake, what exactly does this natural weak factorization system look like in the case of the cone monad? $R$-algebras are bundles equipped with a "coherent" (in what sense?) "global contraction of each of the fibers", but what do $L$-coalgebras look like?

Patrick Nicodemus (Feb 20 2023 at 08:26):

@James Deikun I do not know. Good question. The good news is I have probably* solved this problem of generalizing it to a reasonably general setting, after some blood sweat and tears. But this has taken me a bit away from the central problem of simplicial sets I was discussing either. I will try to answer that question in a bit. My intuition is that this factorization behaves somewhat like the mapping cylinder factorization of a map into a weak equivalence and a cofibration, so I would guess it has similar formal properties as the inclusion of the domain of a map into the mapping cylinder.

Patrick Nicodemus (Feb 20 2023 at 08:26):

*I am still writing up the proof carefully.

dusko (Feb 20 2023 at 11:20):

Patrick Nicodemus said:

Let $\mathcal{C}$ be a category with finite limits. Let $(T,\eta, \mu)$ be a monad on $\mathcal{C}$ which is strong with respect to the Cartesian product by a natural isomorphism $\alpha_{X,Y}: T(X)\times Y\to T(X\times Y)$.

does it help to notice that ${\cal C}/Y$ is the category of coalgebras for the comonad $Y\times(-)$? this "strength" isomorphm then looks like the two distributivities between $T$ and $Y\times (-)$ both ways. one of them should suffice to lift $T$ to the $Y\times (-)$-coalgebras, i.e. ${\cal C}/Y$. (unless i am missing an elephant. it is very late.)

then the other lifting the lccc might be related to what they were doing in SGA4 when they were spelling out the injective-surjective factorizations of geometric morphisms. lifting $T$ would be lifting the covers to the slice...

whether this makes sense or not (sorry that i am not checking myself whether it does), the application to simplicial sets and models looks very interesting...

dusko (Feb 20 2023 at 11:35):

Patrick Nicodemus said:

Let $\mathcal{C}$ be a category with finite limits. Let $(T,\eta, \mu)$ be a monad on $\mathcal{C}$ which is strong with respect to the Cartesian product by a natural isomorphism $\alpha_{X,Y}: T(X)\times Y\to T(X\times Y)$.

maybe you misspoke when you said that $\alpha$ is a natural isomorphism? then $\alpha_{1Y}$ says that $T(Y) \cong T(1)\times Y$. such monads are just multiplications with the monoid $T(1)$. but we don't need $\alpha$ to ve invertible to lift $T$ to ${\cal C}/Y$.
(now really g'nite.)

Patrick Nicodemus (Feb 20 2023 at 23:49):

dusko said:

Patrick Nicodemus said:

Let $\mathcal{C}$ be a category with finite limits. Let $(T,\eta, \mu)$ be a monad on $\mathcal{C}$ which is strong with respect to the Cartesian product by a natural isomorphism $\alpha_{X,Y}: T(X)\times Y\to T(X\times Y)$.

maybe you misspoke when you said that $\alpha$ is a natural isomorphism? then $\alpha_{1Y}$ says that $T(Y) \cong T(1)\times Y$. such monads are just multiplications with the monoid $T(1)$. but we don't need $\alpha$ to ve invertible to lift $T$ to ${\cal C}/Y$.
(now really g'nite.)

I did misspeak. The map $\alpha$ is a natural transformation but not an isomorphism in general, and not in the interesting cases.

Patrick Nicodemus (Jun 19 2023 at 01:28):

This whole thread is me independently reinventing the concept of "simplicial lalis" from this paper by Garner and Bourke. Excellent https://www.sciencedirect.com/science/article/pii/S002240491500170X

Patrick Nicodemus (Jun 19 2023 at 01:29):

Patrick Nicodemus said:

Both of these constructions, when we take $\mathcal{C}$ to be simplicial sets and $T$ to be the cone monad, reduce to the same monad $R$ on $\mathbf{SSet}/Y$, which is a kind of fibered version of the cone monad.

particularly this

Patrick Nicodemus (Jul 17 2023 at 02:11):

Morgan Rogers (he/him) said:

I'm by no means the expert, but this is the first time I've seen someone pulling back along the strength of a strong monad in a construction. For the unit, the coherence condition is that $\alpha_{1,Y} \circ \langle \eta_1 \circ !_X,f \rangle = T(f) \circ \eta_X$; is that necessarily satisfied?

I thought you might be interested in seeing this, as it is closely related to what I mentioned in my comment above.
image.png

Patrick Nicodemus (Jul 17 2023 at 02:11):

https://www.sciencedirect.com/science/article/pii/S002240499600165X

Morgan Rogers (he/him) (Jul 17 2023 at 09:27):

Thanks! It's been long enough that I don't even remember writing that comment, but it's nice to know there is some precedent.