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Write $\mathbf{Cat}$ for the world of categories. Then $\mathbf{Cat}$ has:

  • objects (namely cateories)
  • arrows (namely, functors)
  • proarrows (namely, bimodules)
  • squares (namely, functors between pairs of bimodules)

This makes $\mathbf{Cat}$ into a double category. It also has:

  • an involution denoted $\mathbf{C} \mapsto \mathbf{C}^{op}$ that is covariant on arrows and contravariant on proarrows

For the moment, lets call such a thing an "involutive double category."

Question 0. What are "involutive double categories" actually called?

Now write $\mathbf{Pos}$ for the world of posets. Then $\mathbf{Pos}$ is also an "involutive double category", but it also has the special property that there's at most one square filling any.... ummm.. square. So its kind of "thin", but not at the level of arrows, just at the level of squares.

Question 1. What are these "thin-at-the-level-of-squares involutive double categories" actually called?

Now write $\mathbf{Grpd}$ for the world of groupoids. Then $\mathbf{Grpd}$ is also an "involutive double category." It comes equipped with some additional structure, though, namely, a family of isomorphisms $i_\mathbf{C} : \mathbf{C} \rightarrow \mathbf{C}^{op}$. It seems reasonable to call such a thing a "dagger double category."

Question 2. What are "dagger double categories" actually called?

Now write $\mathbf{Set}$ for the world of sets. Then $\mathbf{Set}$ (with proarrows taken to be relations) is a "thin-at-the-level-of-squares dagger double category."

Question 3. What are such "thin-at-the-level-of-squares dagger double categories" actually called?

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    $\begingroup$ I don't think there is a name. In (higher) category theory giving a name to everything is daunting. $\endgroup$
    – Dimitri Chikhladze
    Commented Feb 27, 2016 at 14:41
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    $\begingroup$ @DimitriChikhladze, sure, but I imagine these things a pretty fundamental, so I hope there's accepted terminology for referring to them. For example, if someone asks you: "what's the difference between arrows and proarrows?" now you can respond: "arrows transform covariantly under involution, proarrows transform contravariantly." This answer only really materializes when you bundle everything up together into "involutive" double categories, so things really are clearer from this viewpoint. $\endgroup$
    – goblin GONE
    Commented Feb 27, 2016 at 14:51

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