1)Category of sets is a (1,0)-category, isn't it?

2)Anafunctors are the factorization of class of functors by equivalence relation(isomorphism) on its values, aren't they?

3)Which kind of category small categories and anafunctors forms? ((2,1)-category?) Let it be $Cat_{ana}$.

4)May I say that $Set$ is "anaisomorphic" to $Set^{op}$ (in $Cat_{ana})$? ( I can not, because of different truthness of proposition "all morphisms to the initial object are isomorphisms". Anafunctors are not related to this statement.)

Is this theory well-developed? What is worth to read about it?

  • $\begingroup$ To see 'all possible values' of a functor $\mathcal A\to\mathcal B$ (such as a limit functor) at once, we can also use its induced profunctor $\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$ with its cograph category which disjointly contain $\mathcal A$ and $\mathcal B$ and 'heteromorphisms' between their objects. Then, the original assignment is given by reflection to $\mathcal B$, so that all target objects are equally presented this way. $\endgroup$ – Berci Sep 4 '16 at 23:10
  • $\begingroup$ @Berci, I am trying to understand what you want to say and which question you answer. Limit functor sends functors of type A->B, where A,B-categories to objects of B, am I right? It sends natural transformations to morphisms between limits in B, am I right? (I didn't understand the rest. I also edited my post by adding answer to 4th question, is it ok?) $\endgroup$ – First Last Sep 5 '16 at 18:13
  • $\begingroup$ Every functor induces two profunctors. (Which are "cocontrabifunctor" and "contracobifunctor" :-) ) $\endgroup$ – First Last Sep 5 '16 at 18:39
  • $\begingroup$ I didn't aim to answer your question, only wanted to reason about why - if not - the theory of anafunctors is not very well developed. Actually, you can choose either profunctor of any given functor. What I meant by cograph is written here: ncatlab.org/nlab/show/…. In the cograph category, the image of an object appears as a [co-]reflection of the object, and if we take all [co-]reflection arrows, we get sth very similar to a saturated anafunctor. $\endgroup$ – Berci Sep 5 '16 at 21:59
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    $\begingroup$ A $(1,0)$-category is just a groupoid, isn't it? (Which sets aren't). $\endgroup$ – neth Sep 6 '16 at 23:54
  1. As stated in the comments, $Set$ is not a groupoid (which is what we mean by (1,0)-category) unless we ask it to be.

  2. I'm not sure what this means, but anafunctors are certain spans of functors, $C\xleftarrow{J} C' \xrightarrow{F} D$, and every functor $G$ gives rise to a canonical functor $C\xleftarrow{id} C \xrightarrow{G} D$. The functor $J$ is surjective on objects and fully faithful, and the set of objects $c'$ of $C'$ in the preimage of a given object $x$ of $C$ give rise to a set of objects $F(c')$ in $D$, all of which are isomorphic, and which Makkai calls the possible values of $F$ at $c$ (think of the limit of a diagram for instance: it is not unique, but all such limits are isomorphic, and any of them will do).

  3. Small categories, anafunctors and transformations form a bicategory (not strict!), which is not a (2,1)-category, unless we throw away all the non-invertible 2-arrows. If you take small groupoids, anafunctors and transformations, then it is a (non-strict) (2,1)-category, purely because all natural transformations involving groupoids are invertible.

  4. Even ignoring the size distinctions (which are not important, but you have been specifying small categories until now), $Set$ is not equivalent to $Set^{op}$ in the bicategory of categories, anafunctors and transformations, as (for instance) $Set$ is a(n elementary) topos and $Set^{op}$ is not, and the property of being a topos is invariant under equivalence (even in the anafunctor setting).

One place to read about anafunctors in a slightly more general setting is in my paper Internal categories, anafunctors and localisations, but the construction goes back to Makkai's paper Avoiding the axiom of choice in general category theory, Journal of Pure and Applied Algebra 108 issue 2 (1996) pp 109-173, https://doi.org/10.1016/0022-4049(95)00029-1, in the case of ordinary categories and to Toby Bartel's thesis Higher Gauge Theory I: 2-Bundles https://arxiv.org/abs/math/0410328, for internal categories. There are more references at the nLab page on anafunctors.


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