Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.     

Weighted limits in simplicial homotopy theory

By combining ideas of homotopical algebra and of enriched category theory, we explain how two classical formulas for homotopy colimits, one arising from the work of Quillen and one arising from the work of Bousfield and Kan, are instances of general formulas for the derived functor of the weighted colimit functor.     

A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a G-space X is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of X can be obtained as a pushout of the categories associated to two open G-subsets covering X. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a G-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.     

The homotopy categorical crossed module of a CW-complex

Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.     

Algebraic models for equivariant homotopy theory over Abelian compact Lie groups

We describe some algebraic models for equivariant rational and p-adic homotopy theory over Abelian compact Lie groups. Received: 12 February 2001; in final form: 15 August 2001 / Published online: 28 February 2002     

Periodic equivariant realK-theories have rational Tate theory

Summary We study a generalized equivariantK-theory introduced by M. Karoubi. We prove, that it is anRO (G, U)-graded cohomology-theory and that the associated Tate spectrum is rational whenG is finite. This implies that for finite groups, the Atiyah-Segal Real equivariantK-theories have rational Tate theory. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag     

The homotopy theory of bialgebras over pairs of operads

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second step we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.     

The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

The homotopy limit problem for Karoubi?s Hermitian K-theory (Karoubi, 1980) [26] was posed by Thomason (1983) [44]. There is a canonical map from algebraic Hermitian K-theory to the Z/2-homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd₂(F[i])<∞, but not in general.     

Geometric versus homotopy theoretic equivariant bordism

By results of Löffler and Comezaña, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G=S¹××S¹, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of Sinha [22]. This is used in order to determine the image of the Pontrjagin-Thom map for toralG.     

An equivariant tensor product on Mackey functors

For all subgroups H of a cyclic p-group G we define norm functors that build a G-Mackey functor from an H-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.     

A stable approach to the equivariant Hopf theorem

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner [S. Waner, Equivariant RO(G)-graded bordism theories, Topology and its Applications 17 (1984) 1-26], the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterised in terms of congruences. This is first shown to be a stable problem, and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe.     

A note on the homotopy theory of pro-G-spectra

This paper represents a step toward a model structure on pro-spectra in which the weak equivalences are the maps inducing pro-isomorphisms of all pro-homotopy groups. We construct a category in which these weak equivalences are inverted and show that we have not inverted “too much,” in the sense that isomorphic objects still give pro-isomorphic cohomology groups.     

Axiomatic homotopy theory for operads

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.     

The 2-dimensional stable homotopy hypothesis

We prove that the homotopy theory of Picard 2-categories is equivalent to that of stable 2-types.     

Transferring homotopy commutative algebraic structures

We show that the sum over planar tree formula of Kontsevich and Soibelman transfers C_∞-structures along a contraction. Applying this result to a cosimplicial commutative algebra A^• over a field of characteristic zero, we exhibit a canonical C_∞-structure on Tot(A^•), which is unital if A^• is; in particular, we obtain a canonical C_∞-structure on the cochain complex of a simplicial set.     

Rational torus-equivariant stable homotopy I: Calculating groups of stable maps

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show that it is of finite injective dimension. It can be used as a model for rational G-spectra in the sense that there is a homology theory

     

Models for homotopy n-types in diagram categories

The classical Mac Lane-Whitehead equivalence showing that crossed modules of groups are algebraic models of connected homotopy 2-types has found a corresponding equivariant version by Moerdijk and Svensson ([22]). In this paper we show that this equivariant result has a higher-dimensional version which gives an equivalence between the homotopy category of diagrams of certain objects indexed by the orbit category of a group H and H-equivariant homotopy n-types for n1.Supported by DGICYT:PS90-0226     

Purity and homotopy theory of coalgebras

We construct a combinatorical monoidal model category on simplicial flat cocommutative coalgebras over a Prüfer domain. The cofibrations are the morphisms which are pure as module maps.