Morphisms in bicatagories - Category theory

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Julian KG

Are the morphisms in a bicatagory always functors. If not what is an example of a bicatagory with morphisms that are not functors?

Asad Saeeduddin

if you take an arbitrary category and introduce some notion of equality of morphisms (even the degenerate one corresponding to the discrete groupoid), you now have a bicategory. if the original morphisms weren't functors, you have an example of what you're looking for

Julian KG

Doesn't this mean you can also have monads that aren't functors?

Asad Saeeduddin

For example you can have monads in the bicategory of profunctors. monads in this bicategory that are equipped with a compatible strength correspond to arrows

Julian KG

But aren't profunctors still functors? Also I'm still new to CT so I'm not sure what you mean by "strength" or exactly what "arrows" refer to in this context (I know what they are in Haskell).

Asad Saeeduddin

Profunctors are still functors, but they're not occurring as morphisms in this bicategory of profunctors in the same way that functors do in the category Cat. Specifically, in Cat a morphism between two categories C and D is a functor C -> D, but in Prof a morphism between categories C and D is a profunctor Cop×DSet\mathbf{C}^{op} \times \mathbf{D} \to \mathbf{Set}. The composition is also not regular functor composition, but is instead a certain notion of "profunctor composition" involving a coend.

Asad Saeeduddin

The arrows I'm referring to are the ones you're familiar with from Haskell. An arrow is a monad in the bicategory Prof\mathbf{Prof} that has a tensorial strength. The Arrow typeclass in Haskell is just talking about such monads on Hask.

You can look up what tensorial strengths and strong monads are, although that's not really relevant if all you're interested in is understanding whether monads can exist in bicategories besides Cat.

Asad Saeeduddin

@Julian KG Here's another example of monads in a bicategory that aren't endofunctors: https://gist.github.com/masaeedu/3e1ba2b7e206837d76343239e0eaa517

Categories are just monads in the bicategory of spans, what's the problem? - catspan.md
Julian KG

I mostly followed this, but a few things tripped me up. First of all spans in category of spans and sets don't seem to be endomorphisms which I thought was a requirement for being a monad. Also if we think about he familiar monad in Cat (as a bicategory) not all morphisms can be monads, only the endofunctors is there any thing that is to spans what functors are to endofunctors?

EDIT: Nevermind I just reread the last paragraph

Asad Saeeduddin

They are endomorphisms, just not endofunctors. A category is a span from some set of objects to itself, with the set of morphisms as the middle set. The "monad" bit is that we have a span morphism from the identity span to the category span, and a span morphism from the squaring of the category span to itself