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Stream: theory: algebraic topology

Topic: adding degeneracies to semi-simplicial sets

Tim Hosgood (Sep 06 2022 at 15:15):

I have a semi-simplicial set that looks like a simplicial set, but writing down the degeneracy operators is a combinatorial headache for which I'm not equipped. If I understand Theorem 5.7 of Rourke and Sanderson's "$\Delta$-sets I: Homotopy Theory" correctly, then I think it's enough for me to show that my semi-simplicial set is Kan, and then just say "add in some degeneracy operators (any choice of which "clearly" gives an equivalent homotopy type, says Kan)".

But this mathoverflow question mentions that the Kan condition for semi-simplicial sets is stronger than the usual one, saying that you also need to "include 1-dimensional horns". What do they mean by this? They say that it means "every vertex is both the source of some 1-simplex and the target of some 1-simplex", but I've never seen the definition of higher-dimensional horns in order to be able to unravel the statement to this.

Tim Hosgood (Sep 06 2022 at 15:17):

In other words, I've "proved" that my semi-simplicial set is Kan in the way that I usually would for simplicial sets: I've shown that all horns lift. What more do I need to do in order to show that my semi-simplicial set is actually Kan (and can thus be completed to a simplicial set)?

Zhen Lin Low (Sep 06 2022 at 15:17):

That sounds like a reasonable definition of 1-dimensional horn to me. (To see this is non-trivial, observe that it is possible to have a semisimplicial set with no 1-cells.)

Tim Hosgood (Sep 06 2022 at 15:19):

sadly I don't have the intuition to know what a reasonable definition would be, but I also can't find it written down anywhere (even in the Rourke and Sanderson paper, which defines the notion of Kan for semi-simplicial sets in, what seems to me, an identical way to usual) — do you have any suggestions for where I might find a reference?

Zhen Lin Low (Sep 06 2022 at 15:21):

I mean, "1-dimensional horns can be filled" means "every 0-cell is the 0th face of some 1-cell and the 1st face of some (possibly different) 1-cell".

Tim Hosgood (Sep 06 2022 at 15:23):

so is a "usual" horn a 0-dimensional horn? I can't figure out the general definition of an n-dimensional horn...

Zhen Lin Low (Sep 06 2022 at 15:24):

In the usual notation, $\Lambda^n_k$ is an $n$-dimensional horn.

Tim Hosgood (Sep 06 2022 at 15:25):

oh, so if I've checked that "all horns fill" (in the usual sense, for simplicial things), then I've already checked for these 1-horns?

Tim Hosgood (Sep 06 2022 at 15:25):

I don't see why the MO question points it out as something extra

Zhen Lin Low (Sep 06 2022 at 15:25):

For a simplicial set, "1-dimensional horns can be filled" is a triviality, because of degeneracies.

Tim Hosgood (Sep 06 2022 at 15:26):

oh I see! thank you!

Tim Hosgood (Sep 17 2022 at 18:45):

is there a relation between this "completion" process (which, I think, is very much not functorial (at least, not "1-categorically", if that makes sense)) which only applies to Kan semi-simplicial sets, and the "free completion" process (given by the left adjoint to the forgetful functor from simplicial sets) which can be applied to any semi-simplicial set?

Amar Hadzihasanovic (Sep 17 2022 at 19:28):

If you freely complete a Kan semi-simplicial set, you do not get a Kan simplicial set; for example you cannot fill horns corresponding to equations $f ; x = \varepsilon$ where $\varepsilon$ is a freely added 1d degenerate simplex.

(But you do still get something which is weakly equivalent to the Kan simplicial set you would get with a choice of degeneracies, because the simplicial/semi-simplicial adjunction is a Quillen equivalence of semi-model categories or weak model categories or whatever works on semi-simplicial sets)

Amar Hadzihasanovic (Sep 17 2022 at 19:37):

I think one good intuition to have is to see a semi-simplicial set as a semigroup-like structure, and a simplicial set as a kind of monoid-like structure, and the left adjoint is like freely adding a unit to a semigroup.

Then a Kan semi-simplicial set is analogous to a semigroup with division, in the sense that all equations $ax = b$ and $xa = b$ have solutions, and a Kan simplicial set is analogous to a monoid with division.
A semigroup with division admits a unique group structure with the same multiplication, so in particular it also becomes a monoid with division, with a (unique) unit $1$.
But if you freely add a unit $\varepsilon$, it is no longer a monoid with division, because, again, you cannot solve $ax = \varepsilon$ or $xa = \varepsilon$.
Still, if you then complete this monoid to a group, it will force $\varepsilon = 1$, and you will get a group isomorphic to the original one.

Tim Hosgood (Sep 18 2022 at 21:12):

Amar Hadzihasanovic said:

(But you do still get something which is weakly equivalent to the Kan simplicial set you would get with a choice of degeneracies [...]

wait, being Kan isn't preserved by weak equivalences?

John Baez (Sep 18 2022 at 21:33):

That would defeat the whole point of weak equivalences, I think. In a model category everything is weakly equivalent to something fibrant.

John Baez (Sep 18 2022 at 21:35):

In the Quillen model structure on simplicial sets, only the Kan complexes are fibrant... but every simplicial set needs to be weakly equivalent to a fibrant one.

Tim Hosgood (Sep 19 2022 at 11:30):

oh woops, that's the sentence I needed to hear to realign my intuition, thanks!

Tim Hosgood (Sep 19 2022 at 11:30):

Amar Hadzihasanovic said:

(But you do still get something which is weakly equivalent to the Kan simplicial set you would get with a choice of degeneracies, because the simplicial/semi-simplicial adjunction is a Quillen equivalence of semi-model categories or weak model categories or whatever works on semi-simplicial sets)

I get this now, thank you!