


Topos Theory / Théorie des topos (Org: Myles Tierney, Rutgers University and/et University of Quebec at Montreal)
 DENISCHARLES CISINSKI, Institut Galilée/Université Paris 13, av. JeanBaptiste
Clément, F93430 Villetaneuse, France
Higher topological Galois theory

Associating to any groupoid its category of representations (i.e. of
presheaves), the 2category of small groupoids can be seen as a full
subcategory of topoi. Moreover, a topos X is Galois (i.e. is the
category of representations of a small groupoid) if and only if it is
locally connected and if any sheaf on X is locally constant. The
classical topological Galois theory can then be stated as follows: The
full inclusions of Galois topoi into locally simply connected topoi
(i.e. locally connected topoi which admits a generating family on which
any locally constant sheaf is constant) has a left adjoint: it is
defined sending a topos X to the topos p_{1} (X) of locally
constant sheaves on X.
If we think of homotopy types as some kind of ¥groupoids
(whatever it means), then there should be some analog of this setting
replacing groupoids by ngroupoids for 0 £ n £ ¥ (and
working with an adequate notion of ntopos). This has been done by
B. Toen in some way for topoi which have the homotopy type of a
CWcomplexe. We shall give another proof of this which allows us to
consider in a unified way an arbitrary n and will make the link
between this point of view and Grothendieck's theory of local test
categories.
 JONATHON FUNK, University of Regina, Regina, SK
Recent results on complete spread geometric morphisms

Complete spread geometric morphisms are canonically equivalent to
Lawvere's topos distributions. Our recent article (Definable
completeness, BungeFunkJibladzeStreicher) identifies a condition
on geometric morphisms that completes a characterization of complete
spread geometric morphisms begun in BungeFunk (Spreads and the
Symmetric Topos, 1996). The condition takes the form of a cover
refinement property expressed in the fibrational theory associated
with a geometric morphism (Moens (1982), Streicher (2003)). This talk
discusses aspects of the condition and the characterization.
Joint work with Marta Bunge, Mamuka Jibladze, and Thomas Streicher.
 ANDRÉ JOYAL, UQAM (Université du Québec à Montréal)
Higher stacks and quasicategories

We recall that the category of simplicial sets admits a model
structure whose fibrant objects are quasicategories [J]. We recall
also that the category of small categories in a Grothendieck topos
admits a Quillen model structure whose fibrant objects are stacks
[T&J]. Here we extend these results to sheaves of
quasicategories. More precisely, we show that the category of
simplicial sheaves admits a Quillen model structure whose fibrant of
objects are higher stacks (higher sheaves of quasicategories).
Joint work with Myles Tierney.
 F. WILLIAM LAWVERE, University at Buffalo, 244 Mathematics Building
The map from Euler reals to Dedekind reals

The rig of uppercuts in Q serves as valuespace for metrics; call it
the Dedekind reals for short (mapping a ring to it would only hit
twosided cuts, but that is a separate issue; if Q denotes the
nonnegative rationals, then the term "arithmetic reals" may be
justified, but for the issue addressed here, Q might as well be
"the constant reals" coming from a lower topos). Euler affirmed that
a real should be determined as a ratio between
infinitesimals. Adopting a rational definition of "ratios", and
conservatively interpreting the appropriate space of infinitesimals
T as the representing object for the tangentbundle functor, we call
Euler reals the part R of the functionspace T^{T} that preserves
the base point. (T is regarded as given as a reflection of physical
experience, and R typically has a unique addition compatible with
the obvious multiplication; if we define D as the part of R of
square 0, the KockLawvere axiom would affirm that there exist units
of time, i.e., isomorphisms T® D, or equivalently certain
nonunique semigroup structures on T itself (in contrast with the
canonical multiplication on our R).) Philosophically, the Euler
reals serve not only to parameterize motion but to provide a basis for
the cause of motion; the cause operates at each single time. By
contrast the Dedekind reals serve to measure by Qapproximations the
result of motion; measuring, like a photograph, kills the particular
motion. Thus the map from Euler reals to Dedekind reals, which is in
urgent need of being understood in any smooth topos of interest, will
therefore not be injective. Any given object in a smooth topos will
induce a function presheaf on finitedimensional varieties; since
continuous functions are not usually smooth, it is unlikely that the
Dedekind reals (even twosided) will be included in R. An inclusion
Q® R of constants is however to be expected, and forms one
ingredient for constructing the map under discussion; the other
ingredient is an ordering on R, inducing in the obvious way the
map from R to parts of Q. Several treatments of SDG postulate this
ordering, but it seems to always turn out that the ordering is not
antisymmetric and that closed intervals are closed under the addition
of infinitesimals, manifesting the noninjectivity of the map. In some
cases there are ways to construct the ordering "synthetically",
i.e., by categorical operations, such as pizero (R) applied
ultimately to the object T.
 JACOB LURIE, Massachusetts Institute of Technology, 77 Massachusetts
Avenue, Cambridge, MA 02139
On InfinityTopoi

The aim of this talk is to introduce the notion of an ¥topos,
which is an ¥categorical analogue of the more classical notion
of a (Grothendieck) topos. Just as an ordinary topos may be thought of
as a "category which looks like the the category of sets", an
¥topos may be thought of an "¥category which looks
like the ¥category of homotopy types". We will explain the
equivalence of two definitions of ¥topoi: one intrinsic, the
other extrinsic. Finally, we show how the theory of ¥topoi may
be used to reformulate certain ideas in classical topology.
 SUSAN NIEFIELD, Union College, Department of Mathematics, Schenectady, NY 12148, USA
Spaces over B and Homotopy in the Topos of Sheaves on B

A topological space X over a base B gives rise to a locale
O(X) over O(B), and hence, an internal locale in the
topos Sh(B) of sheaves on B. The interpretation of concepts in
the internal logic of Sh(B) leads to an internal approach to the
homotopy theory of X over B, including the consideration of a
sheaf of homotopy (bi)groupoids of X over B.
 BOB PARE, Dalhousie University, Halifax, NS B3H 3J5
The Linear Algebra of Categories

In 1984, Joyal and Tierney showed that there was considerable
advantage to study locales as certain monoids in the monoidal category
of sup lattices. Motivated by this, Andy Pitts began the study of
Grothendieck toposes as certain objects in the 2category of
cocomplete categories. In his 1990 thesis, Jonathon Funk conducted a
thorough investigation of this 2category highlighting the deep
parallel between it and the category of modules over a commutative
ring. Since then, this analogy has been exploited to obtain many
interesting results such as, for example, Marta Bunge and Aurelio
Carboni's construction of the symmetric topos. We continue this
study. In particular, we investigate duality and comonoids in the
2category of cocomplete categories.
 DORETTE PRONK, Dalhousie University, Dept. of Math. and Stats, Halifax,
NS B3H 3J5
Hammocks and Free Adjoints, an Equivalence of Localizations

Given a category C with a subcategory W, we discuss
the relationship between the hammock localization L^{H} (C,W), defined by Dwyer and Kan, and P_{2} (C,
W), defined by Dawson, Paré, and the author. The hammock
localization is a simplicial homotopy category which captures the
higher order information implicit in C. The 2category
P_{2} (C, W) is the free 2category obtained by freely
adding right adjoints to the arrows in W. In this talk we
show that the nerves of the hom categories of P_{2} (C,
W) are weakly equivalent to the hom complexes of the hammock
localization, and consequently, that the homotopy categories of the
homcomplexes of the hammock localization are equivalent to the hom
categories of P_{2} (C, W). As a corollary of this
result we obtain several new results for both localizations.
 WALTER THOLEN, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3
Categories of lax algebras as quasitopoi

Depending on a pointed endofunctor of sets and a complete cartesian closed
category as parameters, we form a category of lax algebras and discuss
sufficient (and partly necessary) conditions for this category to be a
quasitopos. This framework leads us to many new and old examples of
quasitopoi.
Joint work with M. M. Clementino and D. Hofmann.

