Category Theory For Beginners: Internal Language of a Topos (Mitchell-Bénabou Language)

In this video we introduce the Mitchell-Benabou Language. Also known as the internal language of a topos. This language allows use to manipulate statements that look like those from set theory and logic, while actually talking about the much more general case of topos theory and topos logic (and topos logic basically looks like intuitionistic logic). In this video we introduce the formal rules behind the Mitchell-Benabou Language. We start by looking at the rules for building terms and interpreting them in the topos. We explain how terms with a type equal to the subobject classifier (formulas) have `extensions' corresponding to the subobjects they classify, and we explain how notation like that of lambda calculus can be used to discuss arrows involving exponential objects. We also discuss term interpretation. Next we apply the Mitchell-Benabou Language to topos logic. We explain how we can use the Mitchell-Benabou Language to say a formula is true, or to say that one formula implies another. We also discuss the idea of sequents (which are claims that formulas imply other formulas), and discuss how we denote chains of implication starting from a given sequent. We use this notation to describe the different term manipulation rules which are sound and complete for topos logic, and justify the rules using our understanding of topos theory. We also go through examples of proofs of statements in topos logic using the manipulation rules we have obtained within the Mitchell-Benabou Language. Here are some videos including further proofs and discussions about particular topics: Implications of implication 1 https://youtu.be/Z4tlsW3HZ_o Understanding False and Not https://youtu.be/dWZ-AT2w7mY Universal Quantification Proof https://youtu.be/6JDEz3MEgJU False in the Mitchell-Benabou Language https://youtu.be/kOZ4hLCQSjo Not in the Mitchell-Benabou Language https://youtu.be/vyEIxvqajHk The proof of Theorem 13.6 (about implication) can be found in this video: Mclarty Topos Basics 1 https://www.youtube.com/watch?v=iuM2Mmi1x7A More information about how "there exists" works can be found here Existential quantifier interpretation https://youtu.be/DJNcZf20p9M The following videos give important practice in using the Mitchell-Benabou Language to formulate important notions: Completing Equalizers with MBL https://youtu.be/bFwESySAwj0 Finding all subobjects containing something https://youtu.be/siytX5wnkIQ Proof that the MBL OR rule works https://youtu.be/OGd9tz7f2iE Proof that the MBL EXISTS rule works https://youtu.be/sdqPK3eKiYg A proof of Theorem 13.9 can be found here "Universal Quantification Proof" https://youtu.be/6JDEz3MEgJU Substitution Lemma argument https://youtu.be/XrumG7RyAmo Substitution rule proof https://youtu.be/XR-8inpOfnU Substitution and equality https://youtu.be/vNe0amN-11c Substitution and equality 2 https://youtu.be/IZk6-XhNNcY Equality, biconditionality, reasoning https://youtu.be/-EIJ9CzdObU (a or b) and (not a) implies b https://youtu.be/JwMhsduAL98 Singleton and comprehension axioms https://youtu.be/GvftlH1A_Oc Axiom of extensionality proof https://youtu.be/yE5FYmNbI6I Product axioms https://youtu.be/9uv_cT6lY00 Correction at 6:06:39 The purple expression at the far bottom left should be the extension of (exists y.Y) phi [rather than just being the extension of phi]. Lots of good applications of MBL are in the following series on "From the MBL to the topos" https://youtu.be/TBUiXvbs2fQ https://youtu.be/wY5cGOfsTGU https://youtu.be/0gSpLSjJX_Q https://youtu.be/HbSBwj0QXYw https://youtu.be/qmNUuCKsKW0 https://youtu.be/3p2k-384mq4 https://youtu.be/kLoUfbZKHVY https://youtu.be/44E_cQbTKyg https://youtu.be/eERlkf--5TA https://youtu.be/TNFnENqM6gA https://youtu.be/y5pDB_JR5yY https://youtu.be/Oat5LeUmUIM https://youtu.be/w_X853zaHgk https://youtu.be/h66t6IlX6d0 https://youtu.be/be_TnFTpYv4 https://youtu.be/qea5do-NCPQ https://youtu.be/oQ8LW4e5XT0 https://youtu.be/ecbhAtafrPI https://youtu.be/eHsHZJvvTNs https://youtu.be/lmRTIErKgYo 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 Power objects and equivalence relations https://youtu.be/_vQ6RNi6_TU