II. Lessons from Homotopy Theory 2. Cohomology and the bar construction A. Quick overview of the bar construction. Cohomology from the bar construction. B. Example of group cohomology: EG and BG. Show that B: Grp -> SimpSet_* is adjoint to pi_1: SimpSet_* -> Grp The "layer-cake" philosophy of cohomology. Show that "two-layer" weak (?) simplicial groups nontrivial only in dimensions 0 and n are classified by group cohomology H^{n+2}(G,A). Detail in the cases n = 0 (central extensions) and n = 1 (2-groups). Relation to Postnikov towers. Examples: central extensions of loop groups, cohomology of Z_2. C. Example of Lie algebra cohomology. Show that "two-layer" semistrict (?) simplicial Lie algebras nontrivial only in dimensions 0 and n are classified by Lie algebra cohomology H^{n+2}(G,A). Detail in the cases n = 0 (central extensions) and n = 1 (Lie 2-algebras). The Whitehead theorems; classifying "semisimple" Lie 2-algebras. The L_infinity operad... how do we functorially get this from the Lie operad? D. Example of quandle cohomology. Relation to group and Lie algebra cohomology...? Higher knot invariants from quandle cohomology??? E. Example of general operad cohomology. Given a linear operad O, construct a simplicial linear operad O_infinity whose algebras in the category of simplicial vector spaces are "weak O-algebras". Show that "two-layer" O_infinity algebras are classified by O-algebra cohomology H^{n+2}(X,A). Note: here we'd need some relation between the bar construction for operads, which gives us O_infinity from O, and the bar construction for algebras of a given operad! Examples: the A_infinity, C_infinity and L_infinity algebras. Note that some of these use permutative operads, others planar operads. F. Deformation theory for operad algebras. Given a linear operad O over a commutative ring k, we can speak of its algebras over the commutative ring k[[x]], or indeed any commutative ring containing k (???). A "deformation" of the O-algebra A over k is an O-algebra A' over k[[x]] which is an extension of A in some sense, "riding" the inclusion k -> k[[x]] These more general extensions should still be classified by some version of the 2nd O-algebra cohomology of A. In short, not only extensions but also deformations fit into the "layer-cake philosophy" of cohomology theory. G. More details on the bar construction. The universal property of the bar construction, a la Trimble.

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