Internal monomorphism in the category of functors

Question

The codensity monad can be seen as a sort of endomorphism monoid.

The internal hom or residual with respect to functor composition as the monoid being basically kan extension.

In Set

$$(G/F)(A) = \forall B, (A \rightarrow F(B)) \rightarrow G(B) $$

This leads to the codensity monad.

I was wondering about what if we required a monomorphism?

I'm not sure how to make this precise but I have a feeling you'd want a condition on monadic bind like

$$ x \mathbin{>>=} f = x \mathbin{>>=} g \rightarrow f = g $$

The codensity monad is a little like the continuation monad and I have a feeling monomorphisms ought to correspond to some sort of scoped or possibly one shot continuation.

I suppose if "codensity monos" are interesting one would for completeness also want to look at codensity epimorphisms and automorphisms.

Answer 1

Let me jot down what -most likely- you have in mind then (re your last comment).

The category $V=[A,A]$ of endofunctors of a category $A$ is strict monoidal with respect to functor composition. The tensor unit is the identity, and all coherence isomorphisms are identities.

What is a category enriched over $V$?

It's a set $C_0$ of objects, plus

• a function $C_0\times C_0\to V$ sending each pair $x,y : C_0$ to an endofunctor $F_{xy} : A \to A$,
• a morphism in $V$ (so, a natural transformation) $u_x : 1_A \Rightarrow F_{xx}$ for every $x : C_0$, and
• a morphism in $V$ (so, a natural transformation)

$$c_{xyz} : F_{xy}\circ F_{yz} \Rightarrow F_{xz}$$

for every $x,y,z : C_0$, such that the composition is associative and $u_x$ serves as unit; writing down a "element-wise" formula for associativity and unitality would be deceiving, because the hom-objects in a $V$-category are functors, not "sets with elements and additional structure".

But you can get some intuition over this definition, if you observe that every $F_{xx}$ is a monad: $u_x$ is its unit, and the multiplication $c_{xxx}=c_x=F_{xx}\circ F_{xx}\Rightarrow F_{xx}$ is the multiplication. So,

a monad on $A$ is just a one-object $[A,A]$-category.

(side remark: this point of view has been adopted to elegantly describe finitary monads as one-object $W$-categories, where $W$ is the category of finitary endofunctors of $\sf Set$; see here)

What about the multiobject version? Well, as far as I can see the multi-object version of a $[A,A]$-category has been studied, but doesn't have a name: monadoid is terrible; maybe paramonad, drawing from this paper?

And what about monomorphisms?

Well, we have a general notion of monomorphism in a $V$-category (for every base of enrichment): see 2.2 here

From each $V$-category $\underline C$ you can extract an ordinary category $C$, with the same objects $C_0$ and where $\hom(x,y) := V(I,\underline{C}(x,y))$ ($I$ the monoidal unit, $\underline{C}(x,y) : V$ the hom-object of $\underline{C}$).

A morphism $m : x\to y$ in the underlying category (so a morphism $m : I\to \underline C(x,y)$ in $V$) now is a $V$-monomorphism when the morphism $C(a,m) : C(a,x)\to C(a,y)$ is a monomorphism in $V$. The definition of $C$ and of the functor $C(a,-)$ now say that $m$ is a monomorphism when

$$ V(I,[a,x]) \xrightarrow{V(I,[a,x]\otimes m)} V(I, [a,x]\otimes[x,y]) \xrightarrow{V(I,c)} V(I, [a,y]) $$

is a monomorphism in $V$.

What does this notion reduce to when $V=[A,A]$?

I am afraid it's going to be a mouthful and not particularly enlightening: the morphisms in the underlying category of a $[A,A]$-category are natural transformations of type $1_A \Rightarrow F_{xy}$; if (building a weird enough $[A,A]$-category where the endofunctor $F_{xy} :A \to A$ can't be pointed for some $x,y : C_0$) there are no such transformations, the hom-set in question is empty. Stupid problems like this besides, though, the notion isn't very enlightening even in case it's not vacuously true: you have a map

$$ [A,A](1_A,F_{ax}) \to [A,A](1_A, F_{ax}\circ F_{xy}) \to [A,A](1_A, F_{ay}) $$ and you want it to be a monomorphism (of sets!)