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 such "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?

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?

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 "thin-at-the-level-of-squares dagger double categories" actually called?

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 such "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?