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Stream: theory: algebraic topology

Topic: name for this variant of simplicial sets

Naso (Dec 10 2021 at 02:55):

Is there any literature about presheaves on $\Delta_a \cup \{\omega\}$, i.e. the category of finite ordinals with the first infinite ordinal $\omega$ adjoined and all order-preserving maps? And is there a name for these?

Alexander Campbell (Dec 10 2021 at 07:58):

@Nasos Evangelou-Oost Note that these are equivalent to $L$-sets, where $L$ is the monoid of all order-preserving endofunctions of the ordinal $\omega$.

Alexander Campbell (Dec 10 2021 at 07:59):

I'm not aware of any literature on these, but @Yuki Maehara gave a talk on a topic very close to this at the Australian Category Seminar on 29 July 2020, under the title "Augmented simplicial sets as $M$-sets".

The $M$ of his title is a certain submonoid (of "eventually consecutive" order-preserving endofunctions) of the monoid $L$ I described above. The main theorem of the talk was that the category of augmented simplicial sets is comonadic over the category of $M$-sets.

Morgan Rogers (he/him) (Dec 10 2021 at 09:37):

Are there slides? Sounds like a potential topological monoid opportunity.

Alexander Campbell (Dec 10 2021 at 19:39):

Morgan Rogers (he/him) said:

Are there slides? Sounds like a potential topological monoid opportunity.

No slides, it was a "board" talk.

Morgan Rogers (he/him) (Dec 11 2021 at 10:54):

In that case, is @Yuki Maehara here occasionally? I want to know more about the adjunction between M-sets and augmented simplicial sets. Specifically, does the left adjoint have any stronger properties?

Yuki Maehara (Dec 11 2021 at 13:26):

I just put my notes on my website: https://yukimaehara.github.io/notes/Msets.pdf
The construction of the left adjoint is quite simple, so if you have any specific property in mind, it shouldn't be too hard to check whether it has that property!

Naso (Dec 12 2021 at 04:43):

Alexander Campbell said:

Nasos Evangelou-Oost Note that these are equivalent to $L$-sets, where $L$ is the monoid of all order-preserving endofunctions of the ordinal $\omega$.

Could you say briefly how this equivalence works please? Is it by extending the adjunction described in Yuki's notes? I.e. just by 'padding' and 'unpadding'?

Todd Trimble (Dec 26 2021 at 18:56):

(Seeing this for the first time; I have a large backlog of unread posts.)

It's that your simplex category with $\omega$ adjoined is the idempotent-splitting completion of $L$ as a one-object category. In general, the category of presheaves $F: C^{op} \to \mathsf{Set}$ is equivalent to the category of presheaves $F': \bar{C}^{op} \to \mathsf{Set}$ on the idempotent-splitting completion (note there is essentially only one way to extend from $F$ to $F'$).