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Jonathan Beardsley (Mar 24 2020 at 23:51):I don't actually have anything to say about this right now other than to hope that this topic heading will make people like me feel a bit more at home.
Jonathan Beardsley (Mar 24 2020 at 23:52):People could use this topic to talk about things like ∞-categories, Quillen model categories, connections to topology, that type of stuff.
Jonathan Beardsley (Mar 24 2020 at 23:52):¯\_(ツ)_/¯
John Baez (Mar 25 2020 at 00:01):Hi! We just had some questions in "applied category theory" from people who want to learn a bit of cohomology but don't want to learn it via algebraic topology. You're at the other end of the spectrum here, pardon the pun.
Jonathan Beardsley (Mar 25 2020 at 00:07):Haha, well if people want to talk about that type of thing, this might be a good topic/stream to talk about it. I'll check back in occasionally, although there are other folks here who know that stuff as well (e.g. you @John Baez!)
Jonathan Beardsley (Mar 26 2020 at 15:14):@nadia esquivel márquez can you see this?
nadia esquivel márquez (Mar 26 2020 at 15:15):@Jonathan Beardsley Yep, thanks
Jonathan Beardsley (Mar 26 2020 at 15:16):As you can see, not much is going on here. We did have a somewhat interesting chat about operads and properads over in #basic questions though.
nadia esquivel márquez (Mar 26 2020 at 15:31):Does anyone know a good place to start reading about reedy categories? (I'm about to start reading Reedy's original text and was wondering about alternatives)
Jonathan Beardsley (Mar 26 2020 at 15:32):Just off the top of my head, have you checked Hirschhorn's book?
Jonathan Beardsley (Mar 26 2020 at 15:32):Page 277 here: https://web.math.rochester.edu/people/faculty/doug/otherpapers/pshmain.pdf
Jonathan Beardsley (Mar 26 2020 at 15:33):(well page 292 of the PDF)
Alex Kavvos (Mar 26 2020 at 16:00):John Baez said:
Hi! We just had some questions in "applied category theory" from people who want to learn a bit of cohomology but don't want to learn it via algebraic topology. You're at the other end of the spectrum here, pardon the pun.
I belong to this class of people as well, I think. Did you have any recommendations?
Alexander Campbell (Mar 26 2020 at 20:03):@nadia esquivel márquez I highly recommend Emily Riehl & Dominic Verity's The theory and practice of Reedy categories, published in TAC.
Mike Shulman (Mar 26 2020 at 20:16):You might also be interested in my preprint https://arxiv.org/abs/1507.01065
Pierre Cagne (Mar 26 2020 at 21:12):Always thought Hovey is good enough to begin with. Mike's paper is great to go further and beyond! If I recall correctly, Reedy never wote about Reedy categories and it was an invention of Kan based on the work of Reedy for simplicial objects in a model category (but I might be mistaken). Kan's paper never got formally published I think, but there are probably copies lying around on the internet (in DJVU form if I should venture a guess).
Mike Shulman (Mar 26 2020 at 21:32):Yes, my understanding is also that Reedy wrote only about simplicial objects and that Kan was the one who formulated the abstract notion of Reedy category.
Tim Hosgood (Apr 14 2020 at 13:45):i know it's categorical homotopy theory instead of homotopical category theory, but i'm hopefully going to be working through some exercises in riehl's book, so good to know that this topic exists so i can ask questions when i get stuck :innocent:
Tim Hosgood (Apr 14 2020 at 13:46):on that note: are these two different things (like algebraic geometry vs geometric algebra), or two names for the same sort of thing (like... well i can't actually think of any other examples)
Morgan Rogers (he/him) (Apr 14 2020 at 14:06):People have studied some model structures on ; I suppose that would be homotopical category theory, but from its very definitions it lies within categorical homotopy theory :stuck_out_tongue_wink: Other interpretations of "homotopical category theory" are possible though.
Thibaut Benjamin (Apr 18 2020 at 19:05):I would have understood "homotopical category theory" as the study of -categories, seen as categories (weakly) enriched over the homotopy types. Or maybe the study of (higher) categories internal to groupoids (e.g. in homotopy type theory) - whereas I would have said categorical homotopy theory for the study of homotopy theory by means of categorical tools
John Baez (Apr 18 2020 at 22:08):So we can use this discussion topic to debate what we should be talking about! :upside_down: