Sharon Hollander
E-Mail: sjh (a) math.ist.utl.pt
Fax: +351-218417035
I am a researcher of the Center for Mathematical Analysis,
Geometry and Dynamical Systems.
Publications
- Geometric criteria for Landweber exactness,
Proc.London Math. Soc. 99 (2009) 697-724.
Abstract:
The purpose of this paper is to give a new presentation of some of the main results
concerning Landweber exactness in the context of the homotopy theory of stacks.
We present two new criteria for Landweber exactness over a flat Hopf algebroid.
The first criterion is used to classify stacks arising from Landweber exact maps
of rings. Using as extra input only Lazard's theorem and Cartier's classification of
p-typical formal group laws, this result is then applied to deduce many of the main
results concerning Landweber exactness in stable homotopy theory and to compute
the Bousfield classes of certain BP-algebra spectra.
The second criterion can be regarded as a generalization of the Landweber exact functor theorem
and we use it to give a proof of the original theorem.
- Diagrams indexed by Grothendieck constructions and stacks on
stacks, which appears in Homotopy,
Homology and Applications 10 (2008) 193-221..
Abstract:
Let I be a small indexing category, G:I^{op} --> Cat
be a functor and BG denote the Grothendieck construction on G.
We define and study Quillen pairs between the
category of diagrams of simplicial sets (resp. categories) indexed on
BG and the category of I-diagrams over N(G) (resp. G). As an
application we obtain a Quillen equivalence between the categories
of presheaves of simplicial sets (resp. groupoids) on a stack M and
presheaves of simplicial sets (resp. groupoids) over M.
-
Realizing modules over the homology of a DGA (with Gustavo
Granja) in
Journal of Pure and Applied Algebra , 212 (2008) 1394-1414.
Abstract:
Let A be a DGA over a field and X a module over H_*(A). Fix an
A_\infty-structure on H_*(A) making it quasi-isomorphic to A.
We construct an equivalence of categories
between A_{n+1}-module structures on X and length n Postnikov systems
in the derived category of A-modules based on the bar resolution of X.
This implies that quasi-isomorphism classes of A_n-structures on X are in bijective
correspondence with the weak equivalence classes of rigidifications of the first n terms of the
bar resolution of X to a complex of A-modules.
The above equivalences of categories are compatible for different values of n which
implies that two obstruction theories for realizing X as the homology of an A-module coincide.
-
Descent for quasi-coherent sheaves on stacks in
Alg. and Geom. Topology
7 (2007) 411-438.
Abstract:
We give a homotopy theoretic characterization of sheaves on a stack and, more generally,
a presheaf of groupoids on an arbitary small site C.
We use this to prove homotopy invariance and generalized descent statements for categories
of sheaves and quasi-coherent sheaves. As a corollary we obtain an alternate proof of a
generalized change of rings theorem of Hovey.
-
Characterizing algebraic stacks in Proc. Amer. Math. Soc.
136 (2008), 1465-1476.
Abstract:
We extend the notion of algebraic stack to
an arbitrary subcanonical site C. If the topology on C is local
on the target and satisfies descent for morphisms, we
show that algebraic stacks are precisely those which are
weakly equivalent to representable presheaves of groupoids
whose domain map is a cover.
-
Hypercovers and simplicial presheaves (with Dan Dugger and Dan
Isaksen)
Math. Proc. Camb. Phil. Soc
136 (2004), 9-51.
Abstract:
Given a Grothendieck site C, Jardine has constructed a model category
structure on simplicial presheaves over C which in some sense captures
the `homotopical sheaf theory' of the site. One element missing from
Jardine's work is a description of the fibrant objects in this model
category, and providing this is one of the goals of the present paper:
we show that they are essentially the simplicial presheaves which
satisfy descent for all hypercovers. Another way of looking at this
result is that we are giving a presentation for the homotopy theory of
simplicial presheaves: it is the universal homotopy theory constructed
from C subject to the relations saying that the homotopy colimit of a
hypercover should be weakly equivalent to the original object. This
is the key observation which lets one construct realization functors
relating the homotopy theory of simplicial presheaves to other model
categories (done elsewhere). In the present paper we also study the
relation between various kinds of descent conditions on simplicial
presheaves.
-
A homotopy theory for stacks in Israel Journal of Math. 163 (2008), 93-124.
Abstract:
We give a homotopy theoretic characterization of stacks on a site C
as the {\it homotopy sheaves} of groupoids on C.
We use this characterization to construct a model category
in which stacks are the fibrant objects.
We compare different definitions of stacks and show
that they lead to Quillen equivalent model categories.
In addition, we show that
these model structures are Quillen equivalent to the
S^2-nullification of Jardine's model structure on sheaves of simplicial sets on C.
- Characterizing Artin Stacks to appear in Math. Zeit.
Abstract:
We study properties of morphisms of stacks in the context of the homotopy theory of
presheaves of groupoids on a small site C.
There is a natural method for extending a property P of morphisms of sheaves on C
to a property P of morphisms of presheaves of groupoids. We prove that the property
P' is homotopy invariant in the local model structure on P(C,Gr)_L when P is
stable under pullback and local on the target. Using the homotopy invariance of the properties
of being a representable morphisms, representable in algebraic spaces, and of being cover
we obtain homotopy theoretic characterizations of algebraic and Artin stacks as
those which are equivalent to simplicial objects in C satisfying certain analogues of
the Kan conditions.
The definition of Artin stack can naturally be extended to a definition of higher order stacks,
which we call n-algebraic stacks, and we provide an analogous characterization of these
in terms of simplicial objects. It is a consequence of this characterization that,
for presheaves of groupoids, n-algebraic is the same as 3-algebraic for all n>2.
As an application of these results we show that a stack is n-algebraic if and only if
the homotopy orbits of a group action on it is.
Misc.
-
PhD Thesis.
This is a version
of my PhD Thesis. It is longer than the above articles and contains
many more details especially about Groupoids and the model structure on them.
Here are
slides
from introductory talks I gave on "Homotopy theory of stacks".
Here are
slides
from the talks I gave on Characterizing algebraic/Artin stacks.
Here are some random
thoughts about economics.