08/10/12 defined subobject classifier, did examples for the category of Sets, Sets^(1+1), groups acting on sets, monoids acting on sets 15/10/12 introduced sieves, proved the bijection between sieves on C and subfunctors of yC, did the subobject classifier in the general case of Sets^(C^op) introduced sheaves (of sets) for topological spaces, partially discussed sheafification 22/10/12 discussed the subobject classifier in simplicial sets (as \n."co-scott-topology on 2^n), discussed a generalization of the subobject classifier for graphs, tried to find the connection. Tried to get it by collapsing standard simplices, calculated the number of antichains in 2^n and concluded that the collapsing will not work for cardinality reasons. 29/10/12 Paolo: reviewed smooth manifolds as a source of examples of bundles (and sheaves), introduced tensor bundles and general vector bundles, briefly mentioned complex manifolds, principal bundles and covering maps. Paolo: continued with germs and sheafification, stated that the sheaf of sections of germs of a sheaf F is isomorphic to F, started a proof. Christian: devised a purely categorical definition of the sheafification functor for $\mathcal{C}$-valued sheafs, where $\mathcal{C}$ is any category with colimits and equalizers. 05/11/12 discussed Christian's idea of defining * : Presheaves -> Sheaves (the Sheafification) as a filtered colimit (works only in some cases) started with Grothendieck topologies: definition of a topology in a small category, definition of a base, some simple lemmas 12/11/12 skipped 19/11/12 Venanzio: repeated Grothendieck topology, base (basis), discussed topology generated from base and vice versa, refinement of bases, bases and topologies for Heyting algebras. Observations: If X is a space, O(X) is a complete Heyting algebra. O(X)^op is a complete lattice, but no Heyting algebra (no exponentials/no infinite distributiv law). Further, a the notions of "complete lattice" and "complete small category" are the same. 26/11/12 Venanzio continued with examples of topologies, defined sheaf over a topology, +-construction with proof 3/12/12 Paolo started discussing elementary topoi 13/12/12 (replacement for 10/12) Paolo continued with elementary topoi, in particular the construction of exponentials [Christmas break] 06/02/13 (?) Venanzio talked about subobjects as a Heyting Algebra 07/02/13 (?) Nicolai: Categorical Semantics of Type Theory 0 11/02/13 Nicolai: Categorical Semantics of Type Theory I, Contextual Categories from universe 18/02/13 Nicolai: Categorical Semantics of Type Theory II - using a universe to get coherence (repetition), structure for 0, 1, identity types