Artin Gluing as Tabulator

General
1

This is a brief note to record an interesting fact that I noticed recently.

Consider the following double category, which I will call \mathbf{Lex}.

  • The objects are logoi (e.g., the categories of sheaves for topoi)
  • The proarrows are left exact functors (functors which preserve finite limits)
  • The arrows are geometric morphisms (e.g. left exact functors with right adjoints)
  • The 2-cells are natural transformations between the squares of left exact functors

It is not hard to see that \mathbf{Lex} has companions; simply forget that a geometric morphism has a right adjoint! What is slightly more surprising (though I believe still known) is that \mathbf{Lex} also has conjoints! The conjoint to a geometric morphism is given by its right adjoint, which preserves all limits and so is in particular lex; the 2-cells in the definition of conjoint are given by the counit and unit of the adjunction!

So \mathbf{Lex} is an equipment. (What I don’t know is whether it is a cartesian equipment, because I’m not so familiar with how coproduct topoi interact with lex morphisms.)

But according to @david.jaz this is all known (I don’t know where it is known). The interesting fact that I think might be original is that Artin gluings are tabulators in \mathbf{Lex}!

Specifically, suppose that we have two logoi E and E', and a lex functor \Phi \colon E \to E'. The Artin gluing is a logos \mathbf{Gl}(\Phi) where an object is given by a tuple (e \in E, e' \in E', f \colon e' \to \Phi(e)). There is then a natural 2-cell

which satisfies the universal property of tabulator! I should give a detailed proof of this, but I should also do other things, so I’m just going to end this here.

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2

I wonder if there’s a general theorem linking commas and tabulators in an equipment? Similarly, the collage (aka cograph) of a profunctor should also be a cotabulator and a cocomma…

3

Is there an abstract definition of comma in a double category? Or I guess is it just what you get from 2-category theory?

4

@owenlynch: I think you would enjoy reading Wood’s 1985 paper “Proarrows II”. One of Wood’s motivating examples of a proarrow equipment (which is equivalent to a double category with companions and conjoints, though not formulated double categorically) is exactly the equipment of toposes and lex functors you describe. §3 of that paper is dedicated to Artin glueing and collages (which are the proarrow equipment theoretic formulation of cotabulators).

5

Thanks @varkor! I had no idea that proarrow equipments went back so far. This is a quite interesting paper. Also, this shows that @david.jaz was right in telling me that “this is all known”!

6

Yes, the tabulator of the composite g_* f^* of the companion of a tight morphism g with the conjoint of a tight morphism f exhibits the comma object f \downarrow g. This is stated, for instance, in Proposition 5.9 of Koudenburg’s On pointwise Kan extensions in double categories.

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