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

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

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).

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 $\mathbf{C}^{op} \times \mathbf{D} \to \mathbf{Set}$. The composition is also not regular functor composition, but is instead a certain notion of "profunctor composition" involving a coend.

The arrows I'm referring to are the ones you're familiar with from Haskell. An arrow is a monad in the bicategory $\mathbf{Prof}$ 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.

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?

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

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

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

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

yes

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

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).

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 $\mathbf{C}^{op} \times \mathbf{D} \to \mathbf{Set}$. The composition is also not regular functor composition, but is instead a certain notion of "profunctor composition" involving a coend.

The arrows I'm referring to are the ones you're familiar with from Haskell. An arrow is a monad in the bicategory $\mathbf{Prof}$ 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.

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

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

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