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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 θ:F→G between homomorphisms B→C is map-valued if θb:Fb→Gb is a map in C for every object b of B.
A cocartesian bicategory is a bicategory B equipped with
Homomorphisms ⊕:B×B→B,E:1→B where 1 is the terminal bicategory,
Map-valued transformations
where Δ:B→B×B is the diagonal homomorphism and !:B→1 is the unique homomorphism,
Invertible modifications
⊕⊕ι→⊕Δ⊕ΔιΔ→Δ⊕ΔEE⋅11→E!E1⊕↓s⇒↓∇⊕1Δ↓t⇐↓Δ∇1E↓u⇒↓ηE⊕→1⊕⊕Δ→1ΔΔI→1EEsatisfying appropriate triangulator (“swallowtail”) coherence conditions.
As ever, there is a lemma that states that if θ is a transformation and f is a map, then the structural 2-cell θ⋅f is an isomorphism. Then, the data above restrict to the bicategory Map(B) whose 1-cells are maps in B, so that ∇,ι,η restrict to strong transformations and ⊕ becomes a left biadjoint to Δ, 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(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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