This is a different approach - more restrictive, I believe - than Mike's work at [[michaelshulman:exact completion of a 2-category]], where he talks about anafunctors in a 2-category (weak, by default, for him). There he talks about anafunctors in the 2-category of 2-congruences in a 2-site, and thinking of them more along the lines as in Cat(S). But I'm very interested in the relation between the two, especially if one could be derived from the other.

One spin-off of this is that I would like to provide another model for the localisation of a 2-category. Here J needs to be weakly cofinal in the class W one wants to invert. One point of my anafunctors paper was to show that the localisation of a 2-category of internal categories had a better model that the default one constructed by Pronk, and this theorem should go through, namely K[W^-1] ~ K_ana. Note that this is (2,2)-category localisation, not (2,1)-category localisation. (As an aside, the approach to localisation via bibundles, which is even simpler to describe, wouldn't work here because that assumes one is in a (2,1)-category.)

The one point which is a bit restrictive is that one needs covers to be an [[ff morphism]] in order to define the bicategory K_ana of anafunctors in K. (This reminds me somewhat of talking about S-local maps in a model theoretic setup, at least when the pretopology J is morally like a cover by open balls or affine schemes. But I haven't thought about this too much yet.)

One direction this may go is if the whole game can be phrased in a suitably 2-categorical way, then perhaps similar techniques could be used to talk about localisation of higher categories (say simplicial categories), at least in special cases. For example, defining weak maps between strict higher categories or something. This is complete speculation, and not a short-term goal by any means.

Thoughts? ]]>

In another thread I came up with a definition of a local isomorphism in a site, working from the definition of a local homeomorphism/diffeomorphism in Top/Diff respectively (with the open cover pretopology in both cases). Then I find that there is a page local isomorphism talking about maps in presheaf categories: such a map is a local isomorphism if becomes an isomorphism on applying the sheafification functor $PSh(S) \to Sh(S,J)$. To quote my definition again

Definition:Let (C,J) be a site (J a pretopology). A map $f:a \to b$ is a J-local isomorphism if there are covering families $(v_i \to b)$ and $(u_j \to a)$ such that for each $u_j$ the restriction $f|u_j$ is an isomorphism onto some $v_i$.

I don’t claim, in the time I have available, to understand the implications of the definition at local isomorphism. I just wonder how it relates to concrete notions like local homeomorphisms (let us work with Top and open covers as covering families). Is a local homeomorphism, after applying Yoneda, a local isomorphism? Does a local isomorphism in the image of Yoneda come from a local homeomorphism? I suspect the answer is yes. Now for the biggie: can a local isomorphism be characterised in terms as basic as my definition as quoted? With my definition one avoids dealing with functor categories (and so size issues, to some extent: $[Top^{op},Set]$ is very big), so if they are equivalent, I’d like to put this somewhere.

Obviously we can take the site in my definition to be a presheaf category with the canonical pretopology or something, and potentially recover the definition at local isomorphism, but for the ease of connecting with geometric ideas, I prefer something simpler.

Any thoughts?

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