```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_*

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

D.  Example of quandle cohomology.  Relation to group and Lie algebra
cohomology...?  Higher knot invariants from quandle cohomology???

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

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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