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Has anyone seen this notion in print? The idea is to capture examples of bicategories like $CoSpan$, $CoRel$, etc.
Here is how one might define cocartesian bicategories, by adapting the development in cartesian bicategory. Following Carboni and Walters, a left adjoint in a bicategory is called a map. Pseudofunctors between bicategories are called homomorphisms. By “transformation”, I mean what Bénabou calls an “oplax transformation” and what Johnstone calls a “lax transformation”. A transformation is strong if its structural 2-cells are invertible. A transformation $\theta: F \to G$ between homomorphisms $\mathbf{B} \to \mathbf{C}$ is map-valued if $\theta b: F b \to G b$ is a map in $\mathbf{C}$ for every object $b$ of $\mathbf{B}$.
A cocartesian bicategory is a bicategory $\mathbf{B}$ equipped with
Homomorphisms $\oplus: \mathbf{B} \times \mathbf{B} \to \mathbf{B}, E: \mathbf{1} \to \mathbf{B}$ where $\mathbf{1}$ is the terminal bicategory,
Map-valued transformations
where $\Delta: \mathbf{B} \to \mathbf{B} \times \mathbf{B}$ is the diagonal homomorphism and $!: \mathbf{B} \to \mathbf{1}$ is the unique homomorphism,
Invertible modifications
$\array{ \oplus & \overset{\oplus \iota}{\to} & \oplus \Delta \oplus & & & & \Delta & \overset{\iota \Delta}{\to} & \Delta \oplus \Delta & & & & E & \overset{E \cdot 1_\mathbf{1}}{\to} & E ! E\\ \mathllap{1_{\oplus}} \downarrow & \overset{s}{\Rightarrow} & \downarrow \mathrlap{\nabla \oplus} & & & & \mathllap{1_{\Delta}} \downarrow & \overset{t}{\Leftarrow} & \downarrow \mathrlap{\Delta \nabla} & & & & \mathllap{1_{E}} \downarrow & \overset{u}{\Rightarrow} & \downarrow \mathrlap{\eta E}\\ \oplus & \underset{1_{\oplus}}{\to} & \oplus & & & & \Delta & \underset{1_{\Delta}}{\to} & \Delta & & & & I & \underset{1_E}{\to} & E }$satisfying appropriate triangulator (“swallowtail”) coherence conditions.
As ever, there is a lemma that states that if $\theta$ is a transformation and $f$ is a map, then the structural 2-cell $\theta \cdot f$ is an isomorphism. Then, the data above restrict to the bicategory $Map(\mathbf{B})$ whose 1-cells are maps in $\mathbf{B}$, so that $\nabla, \iota, \eta$ restrict to strong transformations and $\oplus$ becomes a left biadjoint to $\Delta$, i.e., a 2-coproduct, and $E$ becomes 2-initial ($E$ is for “empty”). The development then proceeds much as it does on the cartesian bicategory page.
Unless I am pretty confused, this notion does not seem to be a simple “co” dual of the notion of cartesian bicategory. For example, these are not co-map valued transformations, and the transformations are still Johnstone-lax, but we are changing directions on the transformations appropriately to get codiagonals and coprojections when we restrict to $Map(\mathbf{B})$ (which will be $Set$ in the examples $CoSpan$ and $CoRel$).
Is it a “co-op” dual?
Ergh. Maybe it’s that simple and I was blind. So I guess the notion of map is co-op self-dual, as is the notion of transformation?
It wouldn’t be the first time I’ve had an Emily Litella moment. :-)
I haven’t checked your definition thoroughly, but I believe that handedness of adjoints and laxity are both co-op self-dual.
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