## Franklin's Notes

### Inclusion subcategory

Note: this note is all about stuff that I've made up myself for purposes of exploration, not read from any authoritative source. Proceed with skepticism.

Let $\mathsf{C}'\subset\mathsf{C}$ be a subcategory of the category $\mathsf{C}$. Let us call $\mathsf{C}'$ an inclusion subcategory if it satisfies the following properties:

1. Every morphism $\iota\in\mathsf{C}'$ is a monomorphism in $\mathsf{C}$
2. Between any two objects $X,Y\in\mathsf{C}'$ there is at most one monomorphism $\iota:X\to Y$ in $\mathsf{C}'$

An equivalent condition to $(2)$ is to require that the following diagram commute for any objects $W,X,Y,Z$ and morphisms $\iota_1,\iota_2,\iota_3,\iota_4$ in $\mathsf{C}'$:

The idea of an inclusion subcategory is my attempt to generalize inclusion maps from $\mathsf{Set}$ to other categories. However, $\mathsf{Set}$ has other inclusion subcategories containing morphisms other than just the inclusion maps. For instance, "classical" inclusion maps of $\mathsf{Set}$ take the form $x\mapsto x$, but we could just as easily consider the subcategory of $\mathsf{Set}$ with morphisms that take the form $x\mapsto {x}$. In this inclusion subcategory of $\mathsf{Set}$, no elements are included in either $\varnothing$ or ${\varnothing}$, making it somewhat similar to a model of set theory with atoms .

This is a consequence of the more general fact that inclusion may not be uniquely/canonically definable. That is, just by looking at the overall structure of a category, there is no way to tell which maps are the inclusion maps. This means that, even for categories like $\mathsf{Set}$ where "inclusion" is defined at a low level, there may be other morphisms that "behave like inclusion" at the higher categorical level.

Here are some basic facts about inclusion subcategories:

• Every inclusion subcategory is skeletal.

• The only isomorphisms in an inclusion subcategory are the identities.

• All automorphisms of an inclusion subcategory are trivial.