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Kevin Carlson (Aug 22 2025 at 22:21):Famously, reflexive coequalizers (or, what's almost the same thing, quotients of reflexive relations) commute with finite but not infinite products in . There are no analogous examples of shapes of diagrams which commute with infinite products other than appropriately filtered categories; in particular, quotients of reflexive relations don't commute with infinite products. This is because you can have reflexive relations for which the shortest path of relators between two points, connected in the quotient, is of unboudned length, and in an infinite product of such relations there is no sufficient length for a path between tuples that are pointwise connected by zigzags of unbounded length. But the long-zigzag problem doesn't occur for quotients of equivalence relations, or indeed for any absolute coequalizers.
Kevin Carlson (Aug 22 2025 at 22:21):Hence my question: do infinite product functors preserve quotients by equivalence relations in ? More generally, does preserve -absolute coequalizers? (It does not reflect them due only, I think, to cases when some factor in the product is empty.)
Kevin Carlson (Aug 22 2025 at 22:46):OK, the equivalence relation case is mentioned by Adámek and friends and is obvious-in-retrospect, but I'm still curious about the absolute coequalizer generalization. It seems like it's going to work so I'm more curious whether this fits into some known story.