Reference for my monads?

Question

I'm looking for a reference for a certain pair of monads on $Cat$. One problem is that I don't know the modern way of thinking about some basic things, so excuse me if my presentation is naive.

First some notation. Let $C$ and $D$ be small categories and let $F\colon C\to Cat\;$ and $G\colon D\to Cat\;$ be functors. Let $(F\Uparrow G)$ denote the following category. $$Ob(F\Uparrow G)=\{(c,d,\ell)\;|\;c\in Ob(C), d\in Ob(D), \ell\colon F(c)\to G(d)\}$$ and whose morphisms are ``natural transformation squares", i.e. $$Hom_{(F\Uparrow G)}((c,d,\ell),(c',d',\ell')=\{(f,g,\alpha)\;|\;f\colon c\to c', g\colon d\to d', \alpha\colon G(g)\circ\ell\to\ell'\circ F(f)\}.$$ With the same setup, let $(F\Downarrow G)\;$ denote the category with the same objects but slightly different morphisms, the only difference being that the natural transformation point the other way: $$Ob(F\Downarrow G)=\{(c,d,\ell)\;|\;c\in Ob(C), d\in Ob(D), \ell\colon F(c)\to G(d)\}$$ $$Hom_{(F\Downarrow G)}((c,d,\ell),(c',d',\ell')=\{(f,g,\beta)\;|\;f\colon c\to c', g\colon d\to d', \beta\colon\ell'\circ F(f)\to G(g)\circ\ell\}.$$

Let $D=\{\star\}$, and denote a functor $G\colon D\to Cat\;$ by $\{G\}$. Now let $C=FCat$, some skeleton of the category of finite categories. Then we have a functors $$(FCat\Uparrow\{-\})\colon Cat\to Cat \;\;\;\;\text{ and }\;\;\;\;(FCat\Downarrow\{-\})\colon Cat\to Cat.$$ I think that each is the functor part of some kind of 2-monad on $Cat$. The unit is "constant" and the multiplication is "Grothendieck construction".

Proving that this is associative, etc, looks laborious, and I don't want to reinvent notation, etc. Is there a good reference for these monads, if they really are monads?

Thanks.

Answer 1

What you are describing is an example of Max Kelly's notion of club, closely connected with the concept of operad. The original references date back to the 70's; one reference is

• G.M.Kelly. On clubs and doctrines. In Category Seminar, Sydney 1972/1973. Springer LNM 420, pp. 181-257 (1974).

Actually, a club is defined to be a monoid in a monoidal category $M$ whose objects are pairs $(C, F: C \to Cat)$ where $C$ is a small category, and where a morphism $(C, F) \to (D, G)$ consists of a functor $H: C \to D$ and a natural transformation $G H \to F$; the monoidal product is a kind of wreath product. There is an "actegory" structure (an action of the monoidal category $M$) on $Cat$,

$$\wr: M \times Cat \to Cat,$$

so that each club = monoid in $M$ induces a monad on $Cat$. You can also find a succinct description of this material here.

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As a reality check, here is a more direct description of the (underlying functor of the) monad on $Cat$, associated with the club structure on the inclusion $i: \mathrm{FCat} \hookrightarrow Cat$, which can be extracted by applying the construction given in Borisov's paper, pp. 3-4. The monad takes a category $D$ to a category which I will denote $\mathrm{FCat} \wr D$ (Borisov uses a semi-direct product symbol). The objects of $\mathrm{FCat} \wr C$ are pairs $(C, l: i(C) \to D)$ where $C \in \mathrm{FCat}$ is a finite category and $l: i(C) \to D$ is a functor. Morphisms $(C, l) \to (C', l')$ of $\mathrm{FCat} \wr D$ are pairs $(F: C \to C', \phi: l \to l' \circ i(F))$ where $\phi$ is a natural transformation.

Unless I have misunderstood something, this gives one of the monads described in the OP. I think the other monad is obtained by a process of dualization (apply $(-)^{op}$, then apply the first monad, then apply $(-)^{op}$ again), so we really only need to worry about the first. As I said, all the laborious technical details were covered long ago in the "Australian school".

Other familiar examples of clubs are where we take the inclusion $i: Set \to Cat$, mapping each set $S$ to the discrete category on $S$; the corresponding monad is the free coproduct completion monad. Another is the composite inclusion $\mathbb{P} \hookrightarrow Set \hookrightarrow Cat$ of the groupoid of finite permutations; here the corresponding monad is the free symmetric (strict) monoidal category construction.