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Stream: learning: questions

Topic: Graded differential cohesive (infinity,1)-toposes

view this post on Zulip Madeleine Birchfield (Mar 21 2025 at 01:54):

Some examples of graded differential cohesive (infinity,1)-toposes that I can think of:

  1. Every differential cohesive (infinity,1)-topos is trivially a graded differential cohesive (infinity,1)-toposes with trivial rheonomic, bosonic, and fermionic modalities (i.e. the modalities are the identity)
  2. The (infinity,1)-topos of super formal smooth infinity-groupoids is a graded differential cohesive (infinity,1)-toposes, according to the work done by Urs Schreiber.

Are there any other examples of graded differential cohesive (infinity,1)-toposes?

view this post on Zulip Morgan Rogers (he/him) (Mar 25 2025 at 09:47):

@Jonas Frey might be interested in this question (although I don't expect him to necessarily have any answers).

view this post on Zulip Jonas Frey (Mar 26 2025 at 11:17):

What do you mean by graded differential cohesive topos? Do you mean what Schreiber calls "solid" topos on the nlab? By "graded" I think of a Z\Z-grading, but I think these have the features of a Z2\Z_2-grading?

view this post on Zulip Madeleine Birchfield (Apr 09 2025 at 23:56):

Jonas Frey said:

What do you mean by graded differential cohesive topos? Do you mean what Schreiber calls "solid" topos on the nlab? By "graded" I think of a Z\Z-grading, but I think these have the features of a Z2\Z_2-grading?

Yeah, that's what I meant; in other parts of nLab it does get called "graded differential cohesive topos" though.

view this post on Zulip Madeleine Birchfield (Apr 09 2025 at 23:57):

Is the Z/2Z\mathbb{Z}/2\mathbb{Z} grading necessary, or can you have solid (∞,1)(\infty,1)-toposes graded by a different ring?

view this post on Zulip Jonas Frey (Apr 10 2025 at 11:39):

@Madeleine Birchfield do you have a link for where the term "graded differential cohesive" is used? I googled it and only found your questions.

view this post on Zulip Jonas Frey (Apr 10 2025 at 11:47):

In my understanding, solid toposes are about the boson/fermion distinction, which is closely related to Z2 grading.

view this post on Zulip David Corfield (Apr 10 2025 at 18:04):

Some discussion of solid cohesion occurs here and then from here.