A Convenient
Homotopy Limit Description
of spaces of
Affine Embeddings
(suspension spectra of)
The pullback of X -f-> Z <-g- Y is {(x,y) | f(x)=g(y)}.
f^*Y
X x_fZg Y
X x_fZg Y  --->  Y
         |                    |
         v                   v
         X  ------->  Z
A category is
a set of object, O, and
a set of morphisms, M
with structure maps
(identity)
i:O -- (c->1_c) --> M
(source)
s:M -- ((c->d)->c) --> O
(target)
t:M -- ((c->d)->d) --> M
(composition)
c:MxM ----> M
A TopCat is
a space of objects, O, and
a space of morphisms, M
with continuous structure maps
i, s, t, c.
If O and M are discrete, we have a usual category
If O is discrete, we have a category enriched over Top
O(C), M(C), C=(O,M)
A functor phi:C_1 --> C_2 between TopCats is
phi_O : O(C_1) --> O(C_2)
phi_M : M(C_1) --> M(C_2)
commuting with the structure maps.
A functor, F:C --> Top, for C discrete is
a space F(c) for each c in O(C)
a morphism F(f):F(c)-->F(d) for each f in M(C)
Could think of F as a single space
F = \coprod_c F(c)
This has a project F --> O, and F(c) is the fiber over c.
Could think of morphisms as acting on F. If x in F(c) and f:c-->d is in M, we obtain F(f)(x) in F(d).
A functor F:C-->Top, for a TopCat C, is a space
F over O(C), along with an action map
F x_O M --> F
satisfying appropriate identities.
If F,G:C-->Top and C is a TopCat, a
natural transformation, eta:F-->G, is
a map over O commuting with the structure maps.
(diagrams)
(diagram)
(diagrams)
(diagrams)
The nerve of a TopCat C is the simplicial space N_*(C) with
(N_i(C))
The geometric realization of C, denoted |C|, is
the realization of the nerve.
If F:C-->D (discrete), the slice category F/d for d in D
is the category with
(objects)
(morphisms)
This is functorial in d, so we have F/- : D --> Cat.
Composing with realization yields |F/-| : D --> Top.
If F is the inclusion of a subcategory, C', we denote the slice category by simply C'/-.
The homotopy limit of F:C --> Top, is
Nat(|C/-|,F)
Tot prod_{N_n} F(c_0)
The homotopy limit of F : C --> Top is
Tot Gamma(N_n(C), t^*F)
(diagram)
If you'd rather,
Gamma(N_n(C), t^*F)=Lambda(N_n(C),F;t)
(If F over O is suitably nice)
(equivalence 1)
(equivalence 2)
If phi : C' --> C is such that
| phi/c | is contractible for all c in C,
Then
(equivalence)
discrete
If phi : C' --> C is such that
|phi/-| --> O(C) is an equivalence,
then
(equivalence)
TopCat
If C' is a sub-TopCat of C, and F is such that
F(c) -~-> holim_{c->c'} F(c')
for all c, then
(equivalence)
This is the right (homotopy) Kan extension of F's restriction to C'.
An affine space, with underlying vector space V, is a set, A, with a difference function, delta:AxA --> V,
satisfying
d(a,b) + d(b,c) + d(c,a) = 0
for all b, d(-,b) is a bijection
write d(a,b) = a-b
An affine space is a vector space that forgot its 0.
You can't add, but you can subtract.
There's a V action: a+v = b if b-a=v.
An affine map, between affine spaces, is a function that commutes with the difference function.
Equivalently, f:A-->B is affine if there is a linear map f_:V(A)-->V(B) such that f(a+v) = f(a)+f_(v).
An affine map V-->W between vector spaces is a linear map followed by a translation.
A space is componentwise affine if it is the disjoint union of affine space.
A map between componentwise affine spaces is affine if it is affine when restricted to each component.
We let Lambda|-S denote that S is a partition (equivalence relation) on S
Lambda|-S is coarser than Lambda' |- S if whenever x==y(L') we also have x==y(L).
Equivalently, equivalence classes (blocks) of L are unions of equivalence classes of L'.
We can take the image of a partition, and for f:S-->T, L |- S, we denote the image partition f(L).
If f : S --> T, we obtain a partition ker f on S by x==y(ker f) iff f(x)=f(y).
If L |-- S and L' |-- S', we say f : S --> S' is a refinement if f(L) <= L'.
Denote the collection of all refinements ref(L, L').
If L |-- S, we say a g : S --> T is non-locally constant if there are x == y (L) such that g(x) != g(y).
Denote the collection of non-locally constant maps nlc(L,T).
Equivalently, g doesn't factor through the quotient S/L.
If f is a refinement, and g is non-locally constant, then gof is non-locally constant.
If A is an affine space with underlying vector space V, and V_0 <= V, we obtain an equivalence relation on A by a == b (V_0) if a-b (= V_0.
We say that a partition of an affine space is affine if it is equivalent to modding out by some linear subspace.
We say that a partition of a componentwise affine space is weakly affine if its restriction to each component is affine.
We say that e |-- A, for componentwise affine A, is rigid if
for all x_1, ..., x_n in some component
with y_1==x_1, ..., y_n==x_n in another component
and t_1, ..., t_n real with sum t_i = 1
we have sum t_i x_i == sum t_i y_i
We say that e |-- A is strongly affine if it is weakly affine and rigid.
An equivalence relation on a componentwise affine space is strongly affine if and only if it is the kernel of an affine map.
A useful technique when dealing with componentwise affine spaces is to pass to the affine sum. If A is componentwise affine, we let A_O denote the affine sum of its components.
One can define an operation which takes an equivalence relation, e, on a componentwise affine space, A, to the finest coarsening by a strongly (weakly) affine partition. We call this operation the closure, and denote it by e- (e~).
The strong closure may be thought of in many ways.
Finest coarseneing by the kernel of an affine map.
Result of an iterative procedure.
Extend e to e^ on A_O, take the weak closure, restrict back to A.
If e |-- A and e' |-- A', we say that an affine f : A --> A' is a refinement if f(e)- <= e'.
The composite of a refinement followed by a non-locally constant map is non-locally constant.
We say e |-- A is complete if it is coarser than ker pi_0
Equivalently, equivalence classes are unions of components.
If d : S --> N_o, and L |-- S, we have the complete partition r = (L |-- S --> N)
with set of components |_|s A^{d(s)} and underlying partition L.
A complete affine partition is trivial if L is trivial and d==0.
Fix m>0, n>=0, and write m_ = {1,...,m}.

Then M = m_ x R^n is the componentwise affine space which has m components, each of which is R^n.
Let M_0 denote the subspace of M consisting of the unit balls.
AffEmb(M,V) denotes the affine maps M --> V which are injective on each component and injective when restricted to M_0.
Let F(m,V) denote the configuration space of m distinct points in V.
That is, F(m,V) = Inj(m_, V)
Let Inj(R^n,V) denote the injective linear maps R^n --> V.
AffEmb(M,V) ~= F(m,V) x Inj(R^n, V)^m
holim Uxu n
holim aU n^m
holim J nu
holim Rxu n
holim aR n^m
holim U|R n^m
Object of C are pairs (r,f) where r is a non-trivial CAP and f : r --> M is non-locally constant.
Morphisms (r,f) --> (r',f') are refinements a : r --> r' such that f = f' o a
(diagram)
This is a TopCat:
O = |_|r nlc(r, M)
M = |_|r,r' ref(r,r') x nlc(r', M)
Let U be the category with
objects: (r,f_0), r a non-trivial CAP, f_0 : r --> m_
morphisms: (r,f_0) --> (r',f_0') are refinements a : r --> r' with f_0 = f_0' o a
Still a TopCat
Define m : U --> Top by m(r,f_0) = nlc_{f_0}(r,M), those non-locally constant maps r --> M which lift f_0.
For a functor F : C --> Top, the category of objects, C x F, has
objects: (c,x) where x in F(c)
morphisms: (c,x) --> (c', x') are morphisms f:c-->c' in C for which F(f)(x) = x'.
C = U x u
For a category, C, the twisted arrow category, a(C) has
Objects: morphisms c --> c' in C
Morphisms: (c-->c') --> (d-->d') are twisted squares
(diagram)
If F, G : C --> Top are contravariant, the assignment
G^F (c-->c') = Maps(F(c'), G(c))
is functorial; G^F : a(C) --> Top,
It's covariant.
holim CxF G ~= holim a(C) G^F
Inside U (objects (r,f_0)) is a subcategory we'll call R, which has objects those (r,f_0) where r is a CAP of the form (L |-- m --> N), and f_0 is the identity on pi_0.
Let J = Rxu, a subcategory of C
An object of J is m affine spaces, A_1, ... A_m, with maps f_i : A_i -> R^n, along with a partition L |-- m so that either L is non-trivial or at least one of the f_i is non-constant.
For a category C, let C^+ denote the category obtained from C by adding a new final object.
If C and D are categories, let C*D, the join, be the full sub-category of C^+ x D^+ of those (c,d) where at least one of c or d is not the final object.
If S^oo X ~= holim_C S^oo F and S^oo Y ~= holim_D S^oo G, then
S^oo (X x Y) ~= holim_C*D F*G
Pf: identify pullbacks: (diagram) and (diagram)
Let P(m) be the category of partitions of m_, ordered by refinement.
Let G(n) be the subcategory of C(R^n) of those (r,f) where r has a single component.
(essentially)
If we think of ^+ as the trivial partition, or the 0 map, we get
J = P * G^{*m}
S^oo F(m,V) ~= holim_P(m) S^oo nlc(-,V)
S^oo Inj(R^n,V) ~= holim_G(n) S^oo nlc(-,V)
Apply the Kan lemma
Apply the slice category lemma
C(M) is functorial with respect to embeddings of M
If M=M(m,n), M'=M(m',n'), and M-->M' is an affine embedding M_0 --> M_0' then C(M) --> C(M')
So M-->M' leads to
holim_C(M) nu <-- holim_C(M') nu
Agreeing with
S^oo Emb(M,V) <-- S^oo Emb(M', V)
Orthogonal Calculus
"Best polynomial approximations" to F : (vector spaces) -->Top (Spectra)
(diagram)
The layers in this tower are homogeneous functors, they can be useful in understanding F.
Excess
For a fixed r, nlc(r,-) is homogeneous of degree e(r).
So we've written Emb as a holim of homogeneous functors
Handy because T_n commutes with holim, and T_n of a homogeneous functor is easy.
We conjecture that T_n S^oo AffEmb(M,V) is the homotopy limit of nu over the subcategory of C(M) consisting of those (r,f) where e(r) <= n.
What about more general manifolds, N? AffEmb(N,V)?
Embedding Calculus: to understand F(N), get a good understanding of F(M) where M ranges over submanifolds of N which are diffeomorphic to disjoint unions of balls.
T_n^O S^oo AffEmb(N,V) ~= T_n^O T_oo^E S^oo AffEmb(N,V)
    ~= T_oo^E T_n^O S^oo AffEmb(N,V)
    ~= holim_{M} T_n^O S^oo AffEmb(M,V)
    ~= holim_{M} holim_{C_k} S^oo AffEmb(m,V)
T_n F(N) = holim_{M} F(M).