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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 1st 2018

    Has anyone seen this notion in print? The idea is to capture examples of bicategories like CoSpanCoSpan, CoRelCoRel, 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:FG\theta: F \to G between homomorphisms BC\mathbf{B} \to \mathbf{C} is map-valued if θb:FbGb\theta b: F b \to G b is a map in C\mathbf{C} for every object bb of B\mathbf{B}.

    A cocartesian bicategory is a bicategory B\mathbf{B} equipped with

    • Homomorphisms :B×BB,E:1B\oplus: \mathbf{B} \times \mathbf{B} \to \mathbf{B}, E: \mathbf{1} \to \mathbf{B} where 1\mathbf{1} is the terminal bicategory,

    • Map-valued transformations

      • :Δ1 B\nabla: \oplus \Delta \to 1_\mathbf{B},
      • ι:1 B×BΔ\iota: 1_{\mathbf{B} \times \mathbf{B}} \to \Delta \oplus,
      • η:E!1 B\eta: E ! \to 1_\mathbf{B},

      where Δ:BB×B\Delta: \mathbf{B} \to \mathbf{B} \times \mathbf{B} is the diagonal homomorphism and !:B1!: \mathbf{B} \to \mathbf{1} is the unique homomorphism,

    • Invertible modifications

      ι Δ Δ ιΔ ΔΔ E E1 1 E!E 1 s 1 Δ t Δ 1 E u ηE 1 Δ 1 Δ Δ I 1 E E\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 θ\theta is a transformation and ff is a map, then the structural 2-cell θf\theta \cdot f is an isomorphism. Then, the data above restrict to the bicategory Map(B)Map(\mathbf{B}) whose 1-cells are maps in B\mathbf{B}, so that ,ι,η\nabla, \iota, \eta restrict to strong transformations and \oplus becomes a left biadjoint to Δ\Delta, i.e., a 2-coproduct, and EE becomes 2-initial (EE 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)Map(\mathbf{B}) (which will be SetSet in the examples CoSpanCoSpan and CoRelCoRel).

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeDec 1st 2018

    Is it a “co-op” dual?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 1st 2018

    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. :-)

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeDec 2nd 2018

    I haven’t checked your definition thoroughly, but I believe that handedness of adjoints and laxity are both co-op self-dual.