Please watch with subtitles. Errata noted in transcript and at bottom of description.
Some content may require a little background in abstract algebra, but there are no topology heavy examples included.
This was originally written for the oral presentation component of my essay module, but the script ended up being way too long. I'd already made the animations, so I've decided to turn it into a crappy video due to the sunk cost fallacy. The audio was recorded in 1-2 takes at 3am, so the quality isn't great (most of the audio is taken from a recording I took purely to time out how long it would take for me to present it). I might update and remake the video in higher quality and in more detail if I have the motivation, but I have too much work right now.
Despite the first subtitle, we only briefly cover the Yoneda lemma in this presentation. Actually, we only briefly cover most of the content in here - I was intending this to be a 15 minute talk, so a lot of material is glossed over.
I am aware I speak quickly - I kept it in mind when recording, but I'll try harder next time. The pacing of transitions is also a bit quick in certain places upon rewatching - I'll make sure to pause more. In the meantime, I have included subtitles which might be helpful if you prefer reading.
Graphics inspired by Oliver Lugg's 27 Unhelpful Facts about Category Theory: https://www.youtube.com/watch?v=H0Ek86IH-3Y.
Main reference during video creation was Basic Category Theory by Leinster and Category Theory in Context by Riehl. Examples of representable functors sourced from notes by Dr. Emanuele Dotto.
Timeline:
00:00 - Introduction
01:08 - Objects
01:40 - Morphisms
02:44 - Compositions
03:01 - Identity
03:22 - Associativity
03:30 - Examples of Categories
06:18 - Product and Dual Categories
07:12 - Duality
07:44 - Commutative Diagrams
08:17 - Isomorphism
09:02 - Functors
10:40 - Covariance and Contravariance
11:15 - Examples of Functors
13:25 - Natural Transformations
15:31 - Vertical Composition
16:53 - Functor Categories
17:18 - Natural Isomorphism
18:22 - Hom Functors
22:19 - Representables
22:40 - Examples of Representables
25:30 - Classifying Spaces
28:19 - The Yoneda Lemma
Errata:
11;13 - "...that a functor is [contravariant], than to...", not "covariant".
18;12 - "...corresponding [objects] are isomorphic...", not "morphisms".
21;57 - the upper string of mappings should be g mapsto hom(h,X)(g) = g o h mapsto hom(B,f)(g o h) = f o (g o h). That is, B and X are the wrong way around in the hom morphisms.
26;59 - "...between the [functions] 1 to R and...", not "functors". (Though, if we treat 1 as the trivial category, and R as a category under ordering, then this does hold for functors in the category of categories. But I really do just mean functions here.)
28;38 - accidentally cut audio, see transcript.
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