Here is an interesting example:

$K_0$ is of course a decategorigication functor. Let us take the version where we mod out by direct sums and not exact sequences- modifications will naturally ensue. This might be thought of as an “extensive decategorication functor” from $A \text{-mod}$ to an abelian group, where direct sums go to sums- sums are the natural decategorication of direct sums.

Broadly, there are two sorts of decategorication functors, and the second is more like a trace. There is an article (here)[https://arxiv.org/pdf/1409.1198.pdf] which defines trace in quite a general way. In brief, we take the abelian group $\oplus_{X \in C} \text{End}_C( X, X)$ and mod out by $fg - gf$.

For instance, take $\mathbb{R} \text{-mod}$. The trace gives $\mathbb{R}$ as an abelian group, along with the canonical map $\text{tr} : \mathbb{R} \text{-mod} \rightarrow \mathbb{R}$, which is identically trace. Viewing $\mathbb{R}$ as a category where morphisms $f : a \rightarrow b$ are elements $c$ such that $c b = a$, we have a contravariant functor, which sends products of morphisms to sums.

It interested me in reading this why there are two sorts of decategorification for abelian categories. But after studying intensive and extensive types some, I wonder, can we view trace as the intensive type corresponding to the extensive type “K_0”?

]]>Aha, so the extensive description in terms of

objects which have purely the negative moment of continuity $\overline{\sharp}$

relates to case such as:

]]>the smooth moduli space of differential n-forms is maximally non-concrete. (dcct, Proposition 1.2.63.)

I brought in that commentary from Science of Logic to this section to see if I can understand how it all links up.

So what is the connection for an object of intensive quantities that

- $X \to \sharp X$ is a monomorphism
- $X$ is a commutative ring object?

finally created *intensive and extensive* with the topos-theoretic formalization following the concise statement in the introduction of *Categories in Continuum Physics*