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Stream: community: general

Topic: categories as monoid objects

Callan McGill (Jun 22 2020 at 22:46):

I seem to recall there is some way to view categories as monoid objects in a (bicategory?) of spans, could anyone enlighten me as to what I am half-remembering?

Evan Patterson (Jun 22 2020 at 22:48):

I learned this from @Christian Williams on Twitter, see this thread.

@ejpatters my favorite monoid is a category - a monoid in Span(Set)(Ob,Ob) this is using that not only in Cat, but in any 2-category K, the category of endomorphisms of an object is monoidal, with tensor given by composition.

- Christian Williams (@c0b1w2)

Nathanael Arkor (Jun 22 2020 at 22:51):

See Section 4.2 of Higher categories, higher operads, and especially Example 4.2.6 for a detailed look at this construction.

Callan McGill (Jun 22 2020 at 22:52):

Thanks both!

John Baez (Jun 22 2020 at 22:53):

It's fun to take the bicategory of spans of sets, look at a monoid object in it, and show it's just a category.

John Baez (Jun 22 2020 at 22:53):

For starters a span of sets is a pair of maps $s: M \to O, t: M \to O'$.

John Baez (Jun 22 2020 at 22:54):

For the span to be a monoid object we need $O = O'$ so we get $s, t: M \to O$. (By the way, a monoid object in a bicategory is usually called a monad.)

John Baez (Jun 22 2020 at 22:54):

The multiplication of the monoid object gives $\circ : M \times_O M \to M$.

John Baez (Jun 22 2020 at 22:55):

From then on it's a downhill slide: $M$ is the set of morphisms of a category, $O$ is the set of objects, $s,t: M \to O$ are the source and target maps, and $\circ$ is composition of morphisms.

Callan McGill (Jun 22 2020 at 23:04):

Cool, thanks! Is there any particular reason you prefer to call that a monad? Is it just because that concept precisely describes the situation where you are dealing with the endomorphism monoidal category in a 2-category?

Nathanael Arkor (Jun 22 2020 at 23:15):

A monad in the 2-category $\mathbf{Cat}$ is precisely a monad on a small category in the traditional sense.

John Baez (Jun 22 2020 at 23:15):

I said it's usually called a monad, not that I prefer anything.

People usually talk about a "monoid object" in a monoidal category, and a "monad" in a bicategory (or 2-category).

John Baez (Jun 22 2020 at 23:16):

A monad in a bicategory is the same thing as a monoid object in the monoidal category of endomorphisms of some object in that bicategory.

John Baez (Jun 22 2020 at 23:19):

It's called a monad "on" that object.

Callan McGill (Jun 22 2020 at 23:52):

Makes sense, thanks!

Rune Haugseng (Jun 23 2020 at 08:25):

You don't get functors as the morphisms between categories from the bicategory of spans though - for that you need to look at the double category of sets, functions, and spans. (Then you can get even more: the double category of (categories, functors, profunctors) arises as a double category of (monoids, monoid morphisms, bimodules) in (sets, functions, spans).)

Mike Shulman (Jun 24 2020 at 05:24):

A different way to get functors and natural transformations is as a free cocompletion under a suitable kind of Kleisli object.