Abstract not found

Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (non-reversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy n-category functor ↑Πn: Smp = n-Cat, left adjoint to a nerve Nn = n-Cat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.

Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphisms.

Introduction The following text arose from the desire to establish a firm relationship between higher order categories and topological spaces. Our approach combines the algebraic features of Michael Batanin's !-operads [1] with the geometric features of Andr'e Joyal's cellular sets [15] and tries to mimick as far as possible the classical construction of the simplicial nerve of a category. Higher order categories have attracted much attention in the last decade, due to their appearance in several mathematical areas. The ultimate goal is perhaps a faithful algebraic description of homotopy systems [13]. Since a homotopy between homotopies has the shape of a disk, the next higher homotopy the shape of a ball, and so on, we chose "ball compexes", i.e. globular sets [26], as the primitive combinatorial objects. The globular structure is precisely what underlies an !-category [26], once its mu

1. Setting up the foundations 3 2. The Eilenberg-Steenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6

this paper appeared four years before Milnor's discovery of exotic dierential structures on spheres [Mil56a]. For an embedding f , he went further and showed that the homotopy type of a tubular neighborhood of f is independent of the dierentiable structure on the ambient manifold. He then introduced the notion of ber homotopy equivalence and proved that the ber homotopy type of the tangent bundle of a manifold is independent of its dierentiable structure. He observed that the 10 J. P. MAY

This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R

Abstract. The categorical concept of Kan extensions form a more general notion of both limits and adjoints. The general definition of Kan extensions is given and motivated by several concrete examples. After providing the necessary background on some basic categorical objects and theorems, the relationship between Kan extensions, limits, and adjoints is expressed through two theorems from [3]. 1. Some Preliminary Categorical Concepts A tremendous array of fields within mathematics draws heavily upon the ideas of limits and adjoints. While these notions are sufficiently general for most uses, there exists a more abstract concept introduced by Kan [2], which encompasses both limits and adjoints. Although the following discussion assumes familiarity with the basic language of category theory, we begin by summarizing some terminology and a few results for reference and clarity. The notion of a limit is the first of these. Limits are easily understood through the auxiliary notion of a cone. Definition 1.1. Given a functor F: D → C, a cone on F is a pair (C, pD) consisting of: • an object C ∈ C, • a morphism pD: C → F D in C, for every object D ∈ D, such that for every morphism d: D → D ′ in D, pD ′ = F d ◦ pD. The name “cone ” is used for a reason; pictorially, cones are situations in which there are morphisms that take the object C to the objects F Di, with the following diagram commuting: F D1 C pD1 ��������� � ������� pD

We present a unified approach to the study of separable and Frobenius algebras. The crucial observation is that both types of algebras are related to the nonlinear equation R 12 R 23 = R 23 R 13 = R 13 R 12, called the FS-equation. Given a solution to the FS-equation satisfying a certain normalizing condition, we can construct a Frobenius algebra or a separable algebra A(R)- the normalizing condition is different in both cases. The main result of this paper is the structure of these two fundamental types of algebras: a finitely generated projective Frobenius or separable k-algebra A is isomorphic to such an A(R). If A is a free k-algebra, then A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B ⊂ A is Frobenius if and only if A has a B-coring structure (A, ∆, ε) such that the comultiplication ∆ : A → A ⊗B A is an A-bimodule map. 0

Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form a cartesian closed subcategory in the topos!Smp of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (non-reversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy n-category functor ↑Πn: Smp = n-Cat, left adjoint to a nerve Nn = n-Cat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.