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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