A convenient
homotopy limit description
of spaces of affine embeddings
AffEmb(M,V)
F(m,V)\times Inj(R^n,V)^m
M=m*R^n
holim_{P(m)}nlc(-,V)
holim_{R^k->R^n} nlc(-,V)
X = holim_C F
Y = holim_D G
X\times Y = holim_{C*D} F*G
AffEmb(M,V)
=
holim_{P*G^{*m}} nu
DONE!
Oh, except, this description isn't "natural".
That's not very convenient
Build a bigger
category!
(more natural)
C(M)
Objects: CAPs
Morphisms: refinements
Complete Affine Partition
Disjoint union of affine spaces
along with a partition of the
set of components
Non-Locally Constant maps
Not injective on some component
- or -
Two equivalent components have disinct images
Refinement
p -> p'
(closure) of image of p is coarser than p'
=
tt