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Tim Hosgood (Aug 22 2020 at 13:38):In http://math.ucr.edu/home/baez/week53.html , @John Baez mentions the idea of -categories, which have -morphisms for all , instead of just for all . Googling these objects proves pretty difficult, so does anybody have any suggestions for places where I can read more about these things?
James Wood (Aug 22 2020 at 16:41):Is this “n-morphisms” in the sense of higher category theory?
Simon Burton (Aug 22 2020 at 21:36):@James Wood Yes...
There's a brief mention of this in the nlab here: https://ncatlab.org/nlab/show/spectrum#combinatorial_spectra
John Baez (Aug 23 2020 at 05:11):The only person I know who is doing serious work on this is Paul Lessard. On Wednesday October 2, 2019 he gave this talk at the Australian Category Seminar:
Speaker: Paul Lessard
Title: Spectra as locally finite Z-groupoids
Abstract: At least since Baez-Dolan '95, we've understood that any correct definition for the notion of symmetric monoidal n-category is equivalent to the one which defines symmetric monoidal n-categories as infinite loop categories. We will synthesize this position with a purely combinatorial treatment of spectra found in Kan '63.
More explicitly, we will extend the proof that the theta categories of Joyal, together with the sets of (higher) spine inclusions provide essentially algebraic presentations of strict categories to provide a definition of the abiding notion of strict Z-categories. We'll then describe a natural subcategory of the category of pointed presheaves on this essentially algebraic theory for Z-categories we'll call the locally finite subcategory and we'll then use Kan's observation to put a model structure on this subcategory which:
- corresponds intuitively to a Z-graded groupoidal composition law; and
- presents the category of spectra.
John Baez (Aug 23 2020 at 05:13):On Wednesday November 9th, 2019, he gave this talk:
Speaker: Paul Lessard
Title: Spectra as Locally Finite Z-groupoids (part 2)
Abstract: In the previous talk we provided a brief development of the objects of stable homotopy theory. In particular, we began with the Freudenthal suspension theorem and argued that homotopy theoretic phenomena split naturally into a low dimensional and a dimension invariant part. We then introduced two Quillen equivalent models for the objects of this dimension invariant part, spectrum objects and Kan's combinatorial spectra.
In this talk, we'll:
develop a just so story for the now disproved Cisinski-Joyal conjecture for (,n)-categories
attempt to convince the audience that the reason the conjecture does not hold is that it was premised on specifically low dimensional intuition.
we'll then explain a grand scheme, inspired by Kan's model, in which higher category theory may be split into a low dimensional and a dimension invariant part; we will introduce a definition for the abiding notion of Z-categories
lastly, we'll show that in this grand scheme, spectra are Quillen equivalent to locally finite Z-groupoids and we'll make some few conjecture about future directions.
John Baez (Aug 23 2020 at 05:16):I see now he has a paper on this subject:
John Baez (Aug 23 2020 at 05:24):I think the subject of -categories is just waiting to be developed; the connection of -categories and spectra is a good start, but there should be a lot more.