Category Theory For Beginners: Adjoint Functors

We start with the homset based definition of an adjunction. We introduce adjoint functors using preordered sets and Galois connections, and hence give examples of adjunctions between monotonic functions. We give an alternative description of adjunctions via universal morphisms, and describe how this works. We describe lots of theory behind adjoint functors, and how they relate to natural transformations like the unit and co-unit. We also give many examples of adjoint functors. In particular we discuss the co-product functor, the diagonal functor, the product functor, the exponential functor, free categories, and adjoint functors related to discrete functors (which relate to connectivity), as well as free/cofree (chaotic) functors for dynamical systems. The later is related to this online question: https://math.stackexchange.com/questions/204554/right-adjoint-to-forgetful-functor-from-dynamical-system-digraph?rq=1 A great guide to the later topics is the second addition of Lawvere's Conceptual Mathematics (although just the first addition seems easily accessible online): https://img.4plebs.org/boards/tg/image/1460/05/1460059215690.pdf A slight correction for time 2:33:56 is that, the exponential functor is just the right adjoint of the functor which does the product of things by a fixed object, whereas the picture gives the impression that the exponential functor is the right adjoint of the functor that does the product when both input objects are variable. In the description of my video Category Theory For Beginners: Yoneda Lemma I started to give links to other unlisted youtube videos I made, mostly following my working through Colin Mclarty's Elementary Categories, Elementary Toposes. I ran out of space in the description, so I put links to the rest of the videos below Extensions and arrows https://youtu.be/eCkt6xcLnIs Initial objects and negation 1 https://youtu.be/F36vYkGoQF4 Initial objects and negation 2 https://youtu.be/7xUNcMizBsY Initial objects and negation 3 https://youtu.be/__jB13bJnzI Coproducts in Toposes 1 https://youtu.be/Kwil3Cvth7Y Coequalizers in Toposes https://youtu.be/IYtY81jm2lQ Fuzzy steps towards the fundamental theorem https://youtu.be/7pqUEGHmZZY First steps in synthetic differential geometry https://youtu.be/AyKYKdtelUw Forgetful and product slice functors https://youtu.be/Yn4NhDPYVbQ More slice functors and the fundamental theorem https://youtu.be/DQ_vGWvoP1k More about the fundamental theorem https://youtu.be/m1lHv5qZNbo philosophical implications of topos theory https://youtu.be/dalJMB4Hi-U complements, decidables, booleans https://youtu.be/2kmCJUq8ZOs External semantics early stage https://youtu.be/iH3iUjQCd9U Fundamentals of topos logic 1 https://youtu.be/iOTZvUmbY8c Fundamentals of topos logic 2 https://youtu.be/uE2h6kv9Ygk Fundamentals of topos logic 3 https://youtu.be/HXiFcw7gDkQ Fundamentals of topos logic 4 https://youtu.be/fzKpvvd8Cb4 Fundamentals of topos logic 5 https://youtu.be/3xPrnRta1_w Even more external semantics https://youtu.be/_MXn1YyC1Hc Categories in Toposes 1 https://youtu.be/hNs3vPOtbQc Categories in Toposes 2 https://youtu.be/JnENI7IL41Q Natural number objects 1 https://youtu.be/GEDl2PkHXBg Natural number objects 2 https://youtu.be/DtQ3ZGODFKg Topos research goals https://youtu.be/-qtya07MRoQ Natural number objects 3 https://youtu.be/qmvJRhBaXus Sets and topologies briefly https://youtu.be/L3ixFT1B1hg Dynamical systems 1 https://youtu.be/XXBN9KdFry4 Dynamical systems 2 https://youtu.be/cyoVVJmnwNM Dynamical systems 3 https://youtu.be/cyoVVJmnwNM Dynamical systems 4 https://youtu.be/52JjPinuJqI Dynamical systems 5 https://youtu.be/dTiT3YjfqkI Dynamical systems 6 https://youtu.be/NUEFDSSszEc Dynamical systems 7 https://youtu.be/NXGwAroSf_c Infinitely rich objects https://youtu.be/loYHVeSuY-w Graph neighborhood problem 1 https://youtu.be/cP1iPAnY7A0 Graph neighborhood problem 2 https://youtu.be/VOFAqOYb7AM Infinitely rich objects 2 https://youtu.be/tKdtB5BLci8 Introducing Heyting algebras https://youtu.be/heOlxlOo9IU Topos research goals 2 https://youtu.be/-Z5V9G6DRus Introducing Heyting algebras 2 https://youtu.be/MKyObwmvgak Introducing Heyting algebras 3 https://youtu.be/vM9aUzUVJgI Relating the algebras with the fundamental theorem https://youtu.be/EZYDl6N18mU