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Stream: theory: algebraic topology

Topic: an endofunctor on simplicial sets

Patrick Nicodemus (Nov 16 2021 at 05:34):

I'm studying a certain adjunction between (augmented!) simplicial sets and another functor category, functors $\Delta\times\Delta^{\rm op}\to\mathbf{Sets}$. (In what follows $\Delta$ always refers to the augmented simplex category).
I want to understand the monad of the adjunction, an endofunctor on simplicial sets; my hope is that it has some kind of nice geometric interpretation beyond the formula for it in terms of ends and coends that I've derived.

The monad sends the terminal object to the coproduct of two copies of the terminal object, it sends the unit interval to three copies of the unit interval, it sends the $n$-simplex to $n+2$ copies of the $n$ simplex. Does anybody have any guesses as to what this might be?

It occurred to me that the $n+1$ simplex has $n+2$ faces in the dimension below the top dimension, I wonder if that's relevant. For example a triangle has 3 sides.

For concreteness the left adjoint sends the presheaf $X$ to the bisimplicial set $K_{q,n} = \int^p X_p\times \operatorname{Hom}(n,p\oplus q)$ (which is covariant in $q$, contravariant in $n$) and the right adjoint sends $K$ to $T(K)$ with $T(K)_p = \int_q K_{q,p\oplus q}$. (The reason i'm interested in this is that for simplicial sets $X, Y$ it's natural to form the "mixed" bisimplicial set $Hom(X_p,Y_q):\Delta\times\Delta^{\rm op}\to \mathbf{Sets}$, and applying the right adjoint $T$ gives the exponential object $Y^X$ (where the monoidal structure is taken to be the Day convolution rather than the Cartesian product)

Would love to hear if anyone has a reference for this.

Morgan Rogers (he/him) (Nov 16 2021 at 10:15):

Patrick Nicodemus said:

It occurred to me that the $n+1$ simplex has $n+2$ faces in the dimension below the top dimension, I wonder if that's relevant. For example a triangle has 3 sides.

Is the image of a 1-simplex under the monad three disjoint 1-simplices or are they connected to form a triangle? That is, is there any substance to this observation? (I'm not very good at reading coends)

Patrick Nicodemus (Nov 16 2021 at 16:13):

Morgan, I believe that they are three disjoint 1-simplices, not connected to form a triangle.

Patrick Nicodemus (Nov 16 2021 at 16:16):

But I find it hard to be certain about these kinds of computations.

Morgan Rogers (he/him) (Nov 17 2021 at 09:44):

If no one answers in the near future, I'll use this as an exercise to learn how to compute coends :rolling_on_the_floor_laughing:

Patrick Nicodemus (Nov 17 2021 at 12:56):

Coends are honestly pretty cool. I find CWM's section on them to be a very straightforward elaboration of their basic properties.
I didn't appreciate how important they were until I started spending more time on enriched category theory in June/July. In the 2-category V-CAT, the unit for the tensor product of V-categories does not generally coincide with the terminal V-category, much like how in an arbitrary monoidal category we don't expect the unit object to be terminal. The upshot is that there is no way to speak about a "constant functor" from C to an object in D -objects in D are in one-to-one correspondence with functors from the unit object 1, but there might not be a canonical choice of functor C -> 1 which makes this work. Even over Ab, for C an Ab-enriched category, there's no reason to expect there to be any additive functor to the one object category 1 with Hom(1,1)=Z, much less a canonical choice of one. (The terminal category 0 has Hom(0,0)=0, the zero abelian group). As a result the entire theory of limits and colimits all just falls apart, one cannot speak of the limit or colimit of a diagram in general. Maybe this all sounds trivial but it was quite striking to me. We have to use these weighted limits/colimits instead which are defined in terms of ends/coends.

Patrick Nicodemus (Nov 17 2021 at 12:57):

But on the bright side the whole theory of weighted limits/colimits is quite elegant. The characterization of the presheaf category as free cocompletion becomes almost definitional from this POV as the presheaf is one of the arguments to the colimit.

Zhen Lin Low (Nov 17 2021 at 12:58):

Personally I would prefer to emphasise weighted (co)limits, for that reason. I find it a pity that they are not covered in CWM.

Patrick Nicodemus (Nov 17 2021 at 12:59):

And only a few weeks ago I read Ross Street's paper on "Yoneda-structures on 2-categories" where he shows that the theory of weighted colimits is more adequately suited for the formal 2-categorical stuff. This really blew me away, and it helped to give a somewhat more concrete reason why the pointwise Kan extensions are so important - they're equivalent to a kind of relative left adjoint, i.e. an "absolute" Kan lift rather than an ordinary Kan lift.

Patrick Nicodemus (Nov 17 2021 at 13:02):

Zhen lin, yes! It is a shame. And still not in the recent textbooks either, I don't think, but I haven't looked. It is somewhat irritating to me that it took me so long to learn the "right definition" of the geometric realization of a simplicial set as the notion of weighted colimit is not so well talked about.

Patrick Nicodemus (Nov 17 2021 at 13:02):

Forget the book by Hatcher, where a "delta complex" is described as handwavily as possible, although it's just the geometric realization of a "semisimplicial" set...

Eric M Downes (Jun 13 2024 at 08:27):

Patrick Nicodemus said:

It is somewhat irritating to me that it took me so long to learn the "right definition" of the geometric realization of a simplicial set as the notion of weighted colimit is not so well talked about.

Perhaps make the definition explicit on nLab?

There is a comment in [[weighted limit]] connecting to [[SimpSet]] where again we have "can be described explicitly on objects as a coend, or as a weighted colimit" ... which I expect no one will find if they aren't already looking.

[[simplicial sets]] Prop 5.1 does mention colimits and coends, but nothing explicitly involving enrichment; maybe that's for the best in terms of readability. Probably a link in parentheses couldn't hurt.

Also the example section of [[weighted colimit]]? It's a very nice example!

I'm not sure what is the best strategy, but perhaps if you ask "in what context would past-me have most found this useful", its not uncommon for nLab to be the second place after textbooks that people look.

John Baez (Jun 13 2024 at 11:37):

Fans of coends may like to see that lots of "realizations" including geometric realizations are given by formulas like

$|X| = \int_{n \in \Delta} X(n) \otimes R(n)$

where $X : \Delta^{\mathrm{op}} \to \mathsf{Set}$ is a simplicial set and $R: \Delta \to C$ is some way of "realizing" simplices in some category $C$ and $\otimes$ is some action of $\mathsf{Set}$ on $C$. We could also replace $\mathsf{Set}$ by something more general.

John Baez (Jun 13 2024 at 11:39):

I don't love coends immensely, so there's a good chance I'm getting this formula a bit wrong, but I've seen many similar-looking formulas, and I think of them all as categorifications of Taylor series

$\sum_{n \in \mathbb{N}} A(n) x^n$

John Baez (Jun 13 2024 at 11:40):

But yes, they are weighted colimits.

Todd Trimble (Jun 13 2024 at 12:06):

I don't love coends immensely, so there's a good chance I'm getting this formula a bit wrong

The only really wrong thing is that the $n$ should be above, not below, on the integral sign.

The tensor symbol is not at all wrong (people do call it "tensoring" after all, the left adjoint to a covariant representable, especially in enriched contexts), but FWIW I tend to use a \cdot symbol instead, to suggest "take a coproduct of $X(n)$ many copies of $R(n)$).

John Baez (Jun 13 2024 at 12:30):

Thanks. Purely by accident it looks like I've drifted closer to the original topic of this thread, which features a coend where the little letter next to the integral sign was in its proper place:

John Baez (Jun 13 2024 at 12:31):

$K_{q,n} = \int^p X_p\times \operatorname{Hom}(n,p\oplus q)$