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

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

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)

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?