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Stream: learning: questions

Topic: relations between test categories

Tim Hosgood (Mar 31 2021 at 14:44):

are there any surveys (or even just nice results) concerning the relations between different test categories? along the lines of how simplices sit nicely inside trees, giving a way of viewing simplicial sets as dendroidal sets. more generally, i was hoping for some big diagram that explains how e.g. globular, cubical, simplicial, dendroidal, Theta, etc. sets all relate to each other, when we have functors between them, how/when/if they're equivalent, and so on

John Baez (Mar 31 2021 at 16:10):

Sounds like a nice idea! I don't know such a survey, but there seem to be lots of results.

Fawzi Hreiki (Mar 31 2021 at 17:02):

While not on test categories in general, there is this paper on the essential subtoposes (aka levels) of cubical and simplicial sets.

Fawzi Hreiki (Mar 31 2021 at 17:04):

Ieke Moerdijk also has some notes on trees which has some stuff on the relationship between simplicial sets and dendroidal sets

Tim Hosgood (Mar 31 2021 at 17:09):

it’s the kind of thing i’d love to write, but unfortunately i only really know about simplicial and globular things!

Tim Hosgood (Mar 31 2021 at 17:10):

but if i can find enough references then it would be a fun task to try to piece them all together into a big picture

Amar Hadzihasanovic (Mar 31 2021 at 18:27):

I'd also be very happy to collaborate on such a survey!

Tim Hosgood (Mar 31 2021 at 20:34):

have the people working on model-independent $(\infty,1)$-category theory got some results in this direction maybe? if there are explicit arguments for why say globular sets and cubical sets both have some notion of "quasi-category" and why these are both equivalent, that would be really nice

John Baez (Mar 31 2021 at 20:48):

I guess the fact that both globular sets and cubical sets are presheaves on generalized Reedy categories might be relevant... very roughly speaking, generalized Reedy categories are categories of shapes where each shape has a "dimension" 0, 1, 2, 3, ...

David Michael Roberts (Mar 31 2021 at 23:51):

Not really an answer, but I wondering something a bit bigger, some years ago: https://mathoverflow.net/questions/24997/is-there-an-interesting-definition-of-a-category-of-test-categories with no real conclusion.

Tim Hosgood (Apr 01 2021 at 16:46):

this is another interesting question indeed