Category Theory For Beginners: Topos Theory And Subobjects

In this video, we introduce topos theory, as a branch of category theory where we generalize notions of inclusion and logic which are traditionally based on set theory. We start by developing ideas of monomorphisms, subobjects, arrow based inclusion and slice categories. Then we introduce the notion of subobject classifiers, firstly according to Lawvere's definition, and then we show this definition is equivalent to the standard pullback based definition. We then illustrate how the subobject classifier works using many examples from the category of sets and the category of dynamical systems. I introduce the idea of a topos, and we explain how to investigate the (potentially non-classical) logical that can come about in a topos, by explaining how to determine the And, Or, Not, and Implies operations in a topos, as well as how to find the intersection and union of subobjects. Further reading recommendations are Conceptual Mathematics: A First Introduction to Categories, by William Lawvere and Stephen Schanuel https://img.4plebs.org/boards/tg/image/1460/05/1460059215690.pdf , and Topoi: The Categorial Analysis of Logic by Ribert Goldblatt. Spivak's lecture notes are also helpful: http://math.mit.edu/~dspivak/teaching/sp18/C7-Logic_of_behavior.pdf Partially erroneous notes that we used to prepare for this video can be found here (but takes these with a large fist of salt !) https://drive.google.com/file/d/1LNBTJHxZYO7vn3uIxRrI8z3ynTu5R5NM/view To see the missing proofs from this video, see this unlisted YouTube video https://youtu.be/qnQVJZuCuwU See this unlisted YouTube video for a description of how to find subobject classifiers in functor categories involving functors into Set. https://youtu.be/ahjkqMcEcKk