Contents

Idea

The generalization of the notion of extensive category from category theory to 2-category theory.

Definition

A 2-coproduct? $A+B$ in a 2-category is said to be disjoint if we have comma squares

The first two say that $A\to A+B$ and $B\to A+B$ are ff and the second two say that they are disjoint subobjects of $A+B$.

A coproduct $A+B$ is said to be universal if for any morphism $Z\to A+B$, the pullbacks

exist and exhibit $Z$ as a coproduct $X+Y$.

Finally, we say a 2-category is extensive if it has finite coproducts which are disjoint and universal. If it also has finite limits we say it is lextensive, and if it is also coherent we call it positive. (Note that disjoint coproducts in a coherent 2-category are always universal.)

An extensive 2-category does satisfy 2-categorical versions of the additional characterizations of an extensive 1-category, but unlike in the 1-categorical case, these alternate conditions do not seem to suffice to characterize extensivity. In particular, though, a 1-category is extensive as a 1-category iff it is so as a homwise-discrete 2-category.

Properties

Preservation

If $K$ is extensive, so is $K^{\mathrm{co}}$, obviously. Less obvious is:

However, the (0,1)-category (= poset) $\mathrm{Sub}(1)$ of (-1)-truncated objects (= subterminal objects) does not inherit extensivity, and in fact posets are almost never extensive: the only disjoint coproduct is $0+0$.

We also have:

References

This is due to

• Mike Shulman, extensive 2-category