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Topos Theory / Théorie des topos
(Org: Myles Tierney, Rutgers University and/et University of Quebec at Montreal)

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DENIS-CHARLES CISINSKI, Institut Galilée/Université Paris 13, av. Jean-Baptiste Clément, F-93430 Villetaneuse, France
Higher topological Galois theory

Associating to any groupoid its category of representations (i.e. of presheaves), the 2-category 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 p₁ (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 n-groupoids for 0 £ n £ ¥ (and working with an adequate notion of n-topos). This has been done by B. Toen in some way for topoi which have the homotopy type of a CW-complexe. 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, Bunge-Funk-Jibladze-Streicher) identifies a condition on geometric morphisms that completes a characterization of complete spread geometric morphisms begun in Bunge-Funk (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 quasi-categories

We recall that the category of simplicial sets admits a model structure whose fibrant objects are quasi-categories [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 quasi-categories. 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 quasi-categories).

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 value-space for metrics; call it the Dedekind reals for short (mapping a ring to it would only hit two-sided 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 tangent-bundle functor, we call Euler reals the part R of the function-space 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 Kock-Lawvere axiom would affirm that there exist units of time, i.e., isomorphisms T® D, or equivalently certain non-unique 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 Q-approximations 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 finite-dimensional varieties; since continuous functions are not usually smooth, it is unlikely that the Dedekind reals (even two-sided) 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 anti-symmetric and that closed intervals are closed under the addition of infinitesimals, manifesting the non-injectivity 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 Infinity-Topoi

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 2-category of cocomplete categories. In his 1990 thesis, Jonathon Funk conducted a thorough investigation of this 2-category 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 2-category 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 P₂ (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 2-category P₂ (C, W) is the free 2-category obtained by freely adding right adjoints to the arrows in W. In this talk we show that the nerves of the hom categories of P₂ (C, W) are weakly equivalent to the hom complexes of the hammock localization, and consequently, that the homotopy categories of the hom-complexes of the hammock localization are equivalent to the hom categories of P₂ (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.