• Lecture 21: Homotopification and higher algebra. Internalizing concepts in categories with finite products. Pushing forward internalized structures using functors that preserve finite products. Why the ‘discrete category on a set’ functor the ‘nerve of a category’ functor and the ‘geometric realization of a simplicial set’ functor preserve products.

• Lecture 22: Monoidal categories. Strict monoidal categories as monoids in or one-object 2-categories. The periodic table of strict -categories. General ‘weak’ monoidal categories.

• Lecture 23: 2-Groups. The periodic table of weak -categories. The stabilization hypothesis. The homotopy hypothesis. Classifying 2-groups with as the group of objects and as the abelian group of automorphisms of the unit object in terms of The Eckmann–Hilton argument.

]]>To see this, choose a Riemannian metric . This is a contractible set of choices and so harmless up to homotopy. Write the symplectic form as for a uniquely defined . Then so that . Hence is positive semi definite. In fact is positive definite because is non degenerate. Now define (e.g. using the functional calculus) . Since everything in sight commutes, you see that , and you have cooked yourself an almost complex structure.

]]>0 –> O –> C^{n+1}(1) –> Q(1) –> 0.

For a complete intersection M of degree d_1, d_2,…d_m in P^n the canonical bundle is O(\sum_i d_i – n -1) restricted to M. This follows from the adjunction formula which just follows by induction and noting that the normal bundle of the smooth zero of a polynomial of degree d is O(d) restricted to the zero. It shows in particular that not all projective varieties are complete intersections as there exist varieties with K^{\dim(M)} = 0.

(written in “asci art Tex” as I keep messing things up the latex/jax thingy).

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Some ideas of geometric quantization extend to and —and even for But I digress… this stuff is sort of strange.

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