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Left adjoints, lenses, and localisation

26th November, 2021

This is really just a cross-post announcement: I wrote a post over on the Topos blog, but it’s something I’ve been thinking about a lot, so I wanted to share a link to it here as well. It’s basically the result of me, knowing a bit about derived categories and model categories, trying to digest this lovely bit of Australian category theory by Bryce Clarke concerning internal lenses. In fact, it’s really a fermented and distilled version of an old blog post from here, namely Cauchy completion and profunctors.

I won’t copy the contents of the entire post here, but I will just repeat the question that I ask at the end (and will refer anybody interested in more context over to my post).

Is there some setting in which I could say something like (maybe swapping “projective” for “injective”) the following: - cofunctors are the morphisms in some “derived” category; - weak equivalences are bijective-on-objects functors; - fibrations are discrete opfibrations; - the category does not have enough projectives; - lenses correspond (via looking at their domain) to K-projective objects.