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