LECTURES ON n-CATEGORIES AND COHOMOLOGY

1. The Basic Principle of Galois Theory 3 1.1. Galois theory and the Erlangen program 3 1.2. The fundamental group 4 1.3. The fundamental groupoid 4 1.4. Eilenberg–Mac Lane Spaces 5 1.5. Klein’s favorite example 6 1.6. Grothendieck’s dream 8 2. The Power of Negative Thinking 10 2.1. Extending the periodic table 10 2.2. The categorical approach 13 2.3. Homotopy n-types 14 2.4. Stuff, structure, and properties 16 2.5. Questions and comments 17 3. Cohomology: The Layer-Cake Philosophy 18 3.1. Factorizations 18 3.2. Cohomology and Postnikov towers 21 3.3. Questions and comments 24 4. A Low-Dimensional Example 27 4.1. Review of Postnikov towers 27 4.2. Example: the classification of 2-groupoids 29 4.3. Relation to the general case 30 4.4. Questions and comments 32 5. Appendix: Posets, Fibers, and n-Topoi 34 5.1. Enrichment and posets 34 5.2. Fibers and fibrations 37 5.3. n-Topoi 41 5.4. Geometric morphisms, classifying topoi, and n-stuff 44 5.5. Monomorphisms and epimorphisms 47 5.6. Pointedness versus connectedness 51 6. Annotated Bibliography 57