Category Theory For Beginners: Topos Theory Essentials

In this video we introduce topos theory in a systematic way, before going for a faster less rigorous tour of some of the deeper ideas in the subject. We start by introducing the idea of a subobject classifier, and how it lets us link ideas about subobjects and logic. We define an elementary topos. We define logical AND, IMPLIES, and FOR ALL. We prove many results interrelating these ideas. Next we discuss other logical notions like FALSE, NOT, and OR. We also discuss epimorphism-monomorphism factorization. Finally, we quickly tour deeper ideas in topos theory, like the idea of forming a Heyting algebra by ordering an object's subobjects by containment. We also discuss notions like the fundamental theorem of topos theory, about how every slice category formed from a topos is also a topos. Here are some more videos exploring more aspects of topos logic: Understanding False and Not https://youtu.be/dWZ-AT2w7mY More results about exponential and power objects https://youtu.be/tErjr1XmRXU "If then" statements in a topos https://youtu.be/z2CHyAJ-mWw Implications of implication 1 https://youtu.be/Z4tlsW3HZ_o Regarding the latter, the result expressing when an arrow w is in [q double-arrow r] , in terms of when [w after [the pullback of q along w]] is in r, can be used together with the Yoneda lemma to determine the D-type elements of [q double-arrow r], in the category of functors from C to Set (for an object D of C). Correction at time 3:32:12 In fact, I do not think (w times 1_A) is always monic, so I should write (w times 1_A) in r, rather than (w times 1_A) contained in r, within Theorem 13.9. A proof of Theorem 13.9 can be found here "Universal Quantification Proof" https://youtu.be/6JDEz3MEgJU Proofs involved in the first part of the video (up until and including the discussion of power objects), can be found in the following four videos: https://youtu.be/iuM2Mmi1x7A https://youtu.be/aC_5PQDQlSg https://youtu.be/TzNDdrlDSrA https://youtu.be/I3qREzStZ3c Here is a link in the description to a video explaining how to construct subobject classifiers and exponential objects for a category of functors from C into Set (for some category C) https://youtu.be/DqmdOZb7v30 Proofs of the results about when arrows are in arrows into power objects can be found here: https://youtu.be/sdNGhUpCb9s An (rough) introduction to the Mitchell-Bénabou language can be found in the following videos (I will probably release a higher quality video on this topic later): https://youtu.be/SmYMPnyZf9U https://youtu.be/ZPerKA2J7DE https://youtu.be/eOZhEwA1slA https://youtu.be/E4FuwWjXFwM https://youtu.be/MCRQJzMGrdM https://youtu.be/DbJVQxBD2JE https://youtu.be/Z2ToNYMuY0w https://youtu.be/FcBeG9PFD1c An attempt to explain how the a partial ordering can be associated with a power object, to form an internal Heyting Algebra can be found here: https://youtu.be/WEuMZapdIXc A very rich set of examples of applications can be found in this series of videos on categories of structured sets: https://youtu.be/Is9vwYUOceg https://youtu.be/c_u-NFr0m7c https://youtu.be/HEurpxoC_98 https://youtu.be/TXr3IxfOhEY https://youtu.be/xBPhH-__dkk https://youtu.be/dLBPYbIEgwA https://youtu.be/kDnUsEIXQHQ https://youtu.be/y6oPlTfY22U https://youtu.be/ZUZe3cHBV7U https://youtu.be/yUFpEAYERkY https://youtu.be/rjAaH0jIITk https://youtu.be/kIY7pAGxHwo https://youtu.be/9yrrtwAZKUY https://youtu.be/TEdk6rFGf8k https://youtu.be/DE4-t5O14Kk https://youtu.be/6RKMw_k_X68 https://youtu.be/1Esdw17yo64 https://youtu.be/mcIrMorIJV4 https://youtu.be/KTNKc1tyhdI https://youtu.be/6zCmbl2yR-s https://youtu.be/F05jntRvM3A https://youtu.be/wQ5ITNBqq18 https://youtu.be/NXtIWMNHIoY https://youtu.be/KsslsRY5Bbc https://youtu.be/1HZh6Vz8PnU https://youtu.be/6qxA2aTH1GM https://youtu.be/2qZWpzaZlFc https://youtu.be/yyFFRxeMbws https://youtu.be/RPB8nH0IZ1E https://youtu.be/4_qL_T1ch_Y https://youtu.be/wWWVfqZMyLQ https://youtu.be/ua5nrJQiosI https://youtu.be/7JvxNcCFl-U https://youtu.be/wP2GoY9Jo1E https://youtu.be/rHqjFsmENBM https://youtu.be/-eGfrlzGMaY https://youtu.be/_U5hIMZu47M https://youtu.be/NurtgTgcnIE https://youtu.be/CGQrXcA7IEs https://youtu.be/Hx7Cd5rJPbk https://youtu.be/c7zKKgLLs2M https://youtu.be/h31Hyo7XdIE https://youtu.be/4k-MpWoakPI https://youtu.be/p59jLhdqcSQ https://youtu.be/YRI2mWRI27M https://youtu.be/HYqZNwhayqs https://youtu.be/TKpCocPb0to https://youtu.be/zWxPVXqypHo https://youtu.be/-fd5x1LWepk https://youtu.be/Zyjrp5CWLAY https://youtu.be/Yc1cq4jVtyY https://youtu.be/LGFNl-5N-sY coequalizer documents and videos https://drive.google.com/open?id=17OON5Bwukl4r-mCYcL8QJlw1E2rQlswS https://youtu.be/pBVVFRdBEno https://youtu.be/asIk1EgGjAE https://youtu.be/HadPH_JW9qA https://youtu.be/I1dQThR51oI Geometric Morphisms https://youtu.be/3L31wrEzL2M https://youtu.be/wYHCG1rLs3I