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Stream: theory: algebraic topology

Topic: adjoint of the construction "graph of a simplicial set"

Leopold Schlicht (Oct 18 2021 at 17:57):

Let $\mathbf{sSet}$ be the category of simplicial sets and $\mathbf{Graph}$ the category of directed multigraphs with morphisms as in Kerodon.001J. There is a functor $F\colon \mathbf{sSet}\to\mathbf{Graph}$, which associates to every simplicial set the graph whose vertices are the 0-simplices and whose edges are the nondegenerate 1-simplices. Does $F$ have a left or right adjoint?

$F$ restricts to an equivalence $\mathbf{sSet}_{\leq 1}\to\mathbf{Graph}$, hence one gets a functor $\mathbf{Graph}\to \mathbf{sSet}_{\leq 1}$. I suspect that the composition $\mathbf{Graph}\to \mathbf{sSet}_{\leq 1}\to\mathbf{sSet}$ is a left adjoint of $F$.

Tim Campion (Oct 18 2021 at 17:58):

yup

Tim Campion (Oct 18 2021 at 17:59):

I guess it depends on what functor you mean by $sSet_{\leq 1} \to sSet$

Tim Campion (Oct 18 2021 at 17:59):

The forgetful functor $sSet \to sSet_{\leq 1}$ has a left adjoint (the inclusion of 1-skeletal simpicial sets) and a right adjoint (the inclusion of 1-coskeletal simplicial sets)

Tim Campion (Oct 18 2021 at 18:00):

there's a bit about that on the nlab

Tim Campion (Oct 18 2021 at 18:00):

The left adjoint and (equivalently) the right adjoint are both fully faithful

Tim Campion (Oct 18 2021 at 18:01):

Oh -- one subtlety

Tim Campion (Oct 18 2021 at 18:01):

$sSet_{\leq 1}$ is equivalent to the category of reflexive directed graphs

Tim Campion (Oct 18 2021 at 18:01):

https://ncatlab.org/nlab/show/reflexive+graph

Tim Campion (Oct 18 2021 at 18:02):

I suppose the forgetful functor from simplicial sets to directed graphs not assumed to be reflexive still has left and right adjoints (given as before by left and right Kan extension) but they are no longer fully faithful

Tim Campion (Oct 18 2021 at 18:03):

the left adjoint will take a graph and freely add a "degenerate" loop to each point.

Leopold Schlicht (Oct 19 2021 at 09:40):

Thanks!

Leopold Schlicht (Oct 19 2021 at 09:48):

Why do you call $\mathbf{sSet}\to \mathbf{sSet}_{\leq 1}$ "forgetful"? I would call its left adjoint $\mathbf{sSet}_{\leq 1}\to\mathbf{sSet}$ forgetful, because it forgets the fact that a simplicial set of dimension $\leq 1$ has dimension $\leq 1$.

Also, the nLab calls the functor $\mathbf{sSet}_{\leq 1}\to\mathbf{sSet}$ "1-skeleton", $\mathrm{sk}_1$. Lurie calls the right adjoint of that functor the "1-skeleton", $\mathrm{sk}_1$.

Leopold Schlicht (Oct 19 2021 at 09:52):

Using the information you gave, to conclude, I would answer my question "Does $F$ have a left or right adjoint?" with "yes, both", because $F\colon\mathbf {sSet}\to\mathbf{Graph}$ factors as $\mathbf{sSet}\to\mathbf{sSet}_{\leq 1}\to\mathbf{Graph}$, where the first functor has both a left adjoint and a right adjoint, as you pointed out, and the second functor is an equivalence. Hence, by composition, one can get both a left and a right adjoint of $F\colon\mathbf {sSet}\to\mathbf{Graph}$.

Leopold Schlicht (Oct 19 2021 at 09:58):

Tim Campion said:

$sSet_{\leq 1}$ is equivalent to the category of reflexive directed graphs

Mhm, here it says that the category of simplicial sets of dimension $\leq 1$ is equivalent to the category of directed multigraphs, without restricting to reflexive graphs.

But one has to define the morphisms between graphs as in Kerodon.001J. So maybe if one doesn't want to define morphisms in that way one has to work with reflexive graphs?

Morgan Rogers (he/him) (Oct 19 2021 at 11:48):

Leopold Schlicht said:

Why do you call $\mathbf{sSet}\to \mathbf{sSet}_{\leq 1}$ "forgetful"?

It forgets all of the (non-degenerate) simplices of dimension >1. As is typical for forgetful functors, this represents a loss of information, whereas the adjoints give canonical ways of extending the information of a graph to a simplicial set.

Leopold Schlicht (Oct 19 2021 at 11:54):

I'm not convinced. Every functor represents a loss of information (at least one can't create information), otherwise one wouldn't be able to specify for each object in the domain category an object of the codomain category.

Usually I think a functor needs to be faithful to be called "forgetful" or "underlying". For instance, I think people would consider it weird if one speaks about the "underlying preorder of a category", even though the preorder associated to a category contains less information than the category.

Reid Barton (Oct 19 2021 at 12:17):

Whether a forgetful functor is faithful depends on what sort of stuff you forget. In this case, we normally think of a simplicial set $X$ as a collection of sets $X_0$, $X_1$, $X_2$, ... with a whole bunch of maps between them (satisfying some relations). The way we obtain the 1-skeleton is to "remember" $X_0$ and $X_1$ and the maps between them, and "forget" everything else. The fact that we're forgetting entire sets $X_2$, $X_3$, ... is what makes the forgetful functor turn out to not be faithful, but it's still forgetful in the ordinary informal sense.

James Deikun (Oct 19 2021 at 12:37):

Which functor is "forgetful" is very much a matter of convention though. There seem to be two conventions which sometimes conflict: use the one that (ironically) forgets least (the fully faithful one, or faithful if there isn't one); and, use the one that is closest to the middle of the adjoint string, breaking ties toward the right.

John Baez (Oct 19 2021 at 13:20):

Leopold Schlicht said:

Tim Campion said:

$sSet_{\leq 1}$ is equivalent to the category of reflexive directed graphs

Mhm, here it says that the category of simplicial sets of dimension $\leq 1$ is equivalent to the category of directed multigraphs, without restricting to reflexive graphs.

But one has to define the morphisms between graphs as in Kerodon.001J. So maybe if one doesn't want to define morphisms in that way one has to work with reflexive graphs?

Yes, it looks like their nasty definition of "morphism between graphs" is the price they pay for not working with reflexive graphs.

John Baez (Oct 19 2021 at 13:23):

In the ordinary category of directed multigraphs, with the usual kind of morphisms, the terminal object has three subobjects. In the category of reflexive graphs it just has two. If you have the patience to sort through Kerodon's nasty definition of morphism, you can see how it works with that.

John Baez (Oct 19 2021 at 13:25):

In the ordinary category of directed multigraphs, which category theorists simply call "graphs", the terminal object is a vertex with an edge from itself to itself. This has three subobjects: itself, the vertex with no edge, and the empty graph.

John Baez (Oct 19 2021 at 13:26):

In the category of reflexive graphs, the terminal object looks the same, but its only subobjects are itself and the empty graph.

John Baez (Oct 19 2021 at 13:28):

In the category $\mathsf{sSet}$, or $\mathsf{sSet}_{\leq 1}$, the terminal object again has just two subobjects.

Leopold Schlicht (Oct 19 2021 at 18:46):

Thanks!

Peiyuan Zhu (Oct 22 2021 at 18:26):

Hi, does anyone has recommendation to understand cohomology structure of information? I think I need to start from the most basic definitions and go from there.

Leopold Schlicht (Oct 25 2021 at 16:24):

Does the inclusion $\mathbf{sSet}_{\leq k}\to \mathbf{sSet}$ have a left adjoint? Its right adjoint is given by $k$-truncation (as discussed above), so $\mathbf{sSet}_{\leq k}$ is coreflective in $\mathbf{sSet}$. I wonder whether it is also reflective.

Morgan Rogers (he/him) (Oct 25 2021 at 16:32):

There's room for confusion here, since there are two canonical functors $\mathbf{sSet}_{\leq k}\to \mathbf{sSet}$, the two adjoints of the $k$-truncation functor $\mathbf{sSet}\to \mathbf{sSet}_{\leq k}$ discussed above; since you specify that you're interested in the left adjoint to the truncation, you're talking about the inclusion of $k$-skeletal simplicial sets.

The inclusion of $k$-simplicial sets fails to preserve products, since products of simplicial sets tend to have non-degenerate higher-dimensional simplices. Is that enough of a hint? As such, it doesn't have a left adjoint.

Reid Barton (Oct 25 2021 at 16:37):

with one exception, $k = 0$!

Leopold Schlicht (Oct 26 2021 at 17:14):

Thanks!

But there's actually no room for confusion, because I said "the inclusion". $\mathbf{sSet}_{\leq k}$ is a (full) subcategory of $\mathbf{sSet}$, and hence there's only one canonical embedding $\mathbf{sSet}_{\leq k}\to\mathbf{sSet}$.

Reid Barton (Oct 26 2021 at 20:30):

Well this is a matter of what definitions you use. $\mathbf{sSet}$ is the category of presheaves on a category $\Delta$, $\mathbf{sSet}_{\le k}$ is the category of presheaves on a full subcategory $\Delta_{\le k} \subseteq \Delta$. The "obvious" functor then is the restriction functor $\mathbf{sSet} \to \mathbf{sSet}_{\le k}$ which takes a functor on $\Delta$ and restricts it to $\Delta_{\le k}$. Any functor of this type has both left and right adjoints. In this particular case, both the left and right adjoints are fully faithful, so they are equivalences between $\mathbf{sSet}_{\le k}$ and full subcategories of $\mathbf{sSet}$--but two different subcategories!

Reid Barton (Oct 26 2021 at 20:30):

Based on your other comments, you must have the left adjoint in mind.

John Baez (Oct 26 2021 at 21:33):

The two adjoints here give the "skeleton" (left adjoint) and "coskeleton" (right adjoint) of a truncated simplicial set:

The $n$-skeleton produces a simplicial set that is freely filled with degenerate simplices above degree $n$. Conversely, the $n$-coskeleton produces a simplicial set having a simplex of degree $m \gt n$ whenever there is a compatible family of $m$-faces.

In simple terms, the skeleton leaves holes, while the coskeleton fills them in.

Leopold Schlicht (Jan 12 2022 at 18:04):

John Baez said:

Yes, it looks like their nasty definition of "morphism between graphs" is the price they pay for not working with reflexive graphs.

Yes, I agree that this definition is pretty nasty. I found a more conceptual way of explaining the equivalence in Kerodon.001N: observe that the usual "category of directed graphs" (the category of presheaves on $\bullet\rightrightarrows\bullet$) can be considered as a coreflective subcategory of the category of semisimplicial sets. This adjunction between directed graphs and semisimplicial sets can be composed with the Teixeira–Barton adjunction between semisimplicial sets and simplicial sets. One gets an adjunction $F\dashv U$ between directed graphs and simplicial sets, which can be restricted to an adjunction $F'\dashv U'$ between directed graphs and simplicial sets of dimension $\leq 1$, since all simplicial sets in the image of $F$ have dimension $\leq 1$. This adjunction $F'\dashv U'$ induces a monad $U'F'$, whose Kleisli category is the category of directed graphs and "nasty" graph morphisms. Since $F'$ is essentially surjective, this category is equivalent to the category of simplicial sets of dimension $\leq 1$.