Terminal coalgebras (Joint work with Tom Leinster)
Eugenia Cheng ( University of Sheffield )
Coalgebras for endofunctors may seem rather lacking in structure at first sight - unlike coalgebras for comonads, they satisfy no axioms. However, terminal coalgebras for endofunctors turn out to be rather interesting. A lemma of Lambek tells us that these are fixed points for the endofunctor in question, and a theorem of Adamek gives us an explicit construction, provided we have enough limits and colimits in our ambient category. One consequence is that we have a way of constructing infinite versions of algebraic structures that are usually constructed by induction and thus can usually only be finite. Examples include infinite words, infinite trees and strict $\omega$-categories. In this talk we will present the basic theory and examples, and also give a new example: an $\omega$-dimensional version of Trimble's weak $n$-categories. This talk will not assume knowledge of category theory beyond categories and functors, although familiarity with limits in categories will help.