# 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.

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