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I've never really thought much about distributivity of limits and colimits -- I tend to think more about commutativity of limits and colimits. This question makes me want to change that.

The question of which limits commute with which colimits in the category of sets has a complicated answer, but important special cases which are not so complicated. This leads to the title question:

Question:

  1. Which limits distribute over which colimits in the 1-category $Set$?

  2. Which limits distribute over which colimits in the $\infty$-category $Spaces$?

Is it complicated like with commutativity? Or is it maybe simpler? If it's complicated, are there good simplifying assumptions one can make?

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    $\begingroup$ Does the general definition of distribution of limits and colimits appear in the literature, or just on the nlab? $\endgroup$
    – Tim Campion
    Commented Feb 9, 2021 at 15:33
  • $\begingroup$ I’m looking for something similar. I am currently trying to understand the general nLab statement for what distributivity would mean if K is a functor out of I, but then there is the second question of am I even in the context where such distributivity holds! $\endgroup$
    – cheyne
    Commented Apr 24, 2021 at 15:28
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    $\begingroup$ @cheyne At this point I've talked to quite a few people about this and everyone has the same reaction: the notion on the nlab seems very interesting, but frustrating and difficult to actually get one's head around and seems to be kind of an "orphan" -- no literature discussing the topic. One place where this notion is discussed is around 7.11 in Chu and Haugseng; the discussion there was inspired by the nlab apparently. $\endgroup$
    – Tim Campion
    Commented Apr 24, 2021 at 15:32

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