Factorization Systems on Posets -
The sometimes-confusing distinction between codomain (aka range or target) and image is nicely formalized by the notion of factorization system from category theory: every function factors (in an essentially unique way) as a surjection onto its image followed by the inclusion of image into codomain. This is the only interesting factorization system on the category of sets and functions, but such systems proliferate when we consider other categories. In this talk, I will motivate categories and factorization systems, and then expose some of the surprising and deep combinatorics of factorization systems on posets (i.e., partially ordered sets). In an unexpected turn, when applied to subgroup lattices, these results have important applications to equivariant infinite loop space theory. Much of this work is joint with Reed students and colleagues, who will be extensively praised/embarrassed during the talk.