An Almost Slice Category

I just wanted to get this definition down in case I want to think about it later.

Suppose we have a category $\C$ with finite products. Suppose we have a monoid object $(R, \square : R \x R \to R, e : 1 \to R)$, which has a monoid action $\cdot : R \x M \to M$ on an object $M$.

With this equipment, we can define another category, which is sort of like the slice $\C / M$. Its objects are the same as $\C / M$: they're maps $f : A \to M$ into $M$. Now to be a morphism from $f : A \to M$ to $g : B \to M$ you need to supply a $k : A \to B$ but also a $\phi : A \to R$ such that the following diagram commutes:

The identity map from $f$ to itself is given by

which commutes by monoid action laws. Composition works according to the following diagram chase:

Zero morphisms

Suppose we have a subset $Z$ of all of the morphisms into $R$, thought of as morphisms that are "close enough to the identity". If it satisfies

$e \in Z$
If $\psi \in Z$ then $\psi k \in Z$
If $\phi \in Z$ and $\psi \in Z$, then $\phi \square \psi\in Z$

Then the subset of morphisms of the above category where $\phi \in Z$ is a subcategory.

A Monadic Generalization

Suppose we have a monad $T : \C \to \C$ and a monad algebra $\cdot : TM \to M$. Suppose for each $A$ there is subset of the morphisms $A \to TA$ that are designated "$T$-boring". They are required to satisfy the following axioms:

$\eta_A$ is $T$-boring.
If $f, g : A \to TA$ are $T$-boring, then their kleisli composition $\mu_A \o Tg \o f$ is.
If $\psi : B \to TB$ is $T$-boring, and $k : A \to B$ is any map, then there exists a $T$-boring map $\phi : A \to TA$ such that $\phi \o k = Tk \o \psi$.

Then we define a category whose objects are maps into $M$, and whose morphisms from $f : A \to M$ to $g : B \to M$ consist of a map $k : A \to B$ and a $T$-boring map $\phi : A \to TA$ such that $g \o k = a\o Tf \o \phi$. Identities and composition work by the following diagrams:

Dinatural Generalization

Suppose as before that we have a monad $T : \C \to \C$ and $\cdot : TM \to M$. Zeroness is represented by a monoid-valued presheaf $Z : \C^\op \to \rmon$. Define $Z' : \C^\op \x \C \to \rset$ by \[Z'(C_1, C_2) = U(Z(C_1))\] for the forgetful $U : \rmon \to \rset$. Require a dinatural transformation \[\alpha : Z' \to \hom(\dash, T\dash)\] For each $A$, compatibility of the monoid structure $(*_A, e_A)$ on $Z(A)$ with the monad structure is given by: \[\alpha_A (e_A) = \eta_A\] \[\alpha_A(x*_A y) = \mu_A \o T\alpha_A(x) \o \alpha_A(y)\]

The Special Case I Thought Was Interesting

Let $R$ be a commutative semiring. Let $R^\pm$ be the commutative semiring whose underlying set is $R \x R$ and where multiplication is given by \[(a,b) (c, d) = (ac+bd, ad+bc)\] to simulate negatives. Let $- : R^\pm \to R^\pm$ be defined by swapping components. Let $R^i$ be the commutative semiring whose underlying set is $R^\pm \x R^\pm$ and where multiplication is given by \[(a,b) (c, d) = (ac-bd, ad+bc)\] to simulate complex numbers. There is a homomorphism of the multiplicative groups $R^i \to R^\pm$ given by squared norm: \[\| (a, b) \|^2 = a^2 + b^2\] Although we don't get commutativity on the nose:

we can nonetheless achieve:

Which is kind of like establishing a lax 2-cell across the first square?