Homology and cohomology via enriched bifunctors

@article{Shimakawa2010HomologyAC,
  title={Homology and cohomology via enriched bifunctors},
  author={Kazuhisa Shimakawa and Ken'ichi Yoshida and Tadayuki Haraguchi},
  journal={arXiv: Algebraic Topology},
  year={2010}
}
We show that the category of numerically generated pointed spaces is complete, cocomplete, and monoidally closed with respect to the smash product, and then utilize these features to establish a simple but flexible method for constructing generalized homology and cohomology theories by using the notion of enriched bifunctors. 
STEENROD-ČECH HOMOLOGY-COHOMOLOGY THEORIES ASSOCIATED WITH BIVARIANT FUNCTORS
Let NG0 denote the category of all pointed numerically generated spaces and continuous maps preserving base-points. In [SYH], we described a passage from bivariant functors NG 0 ×NG0 → NG0 to
Homotopy structures of smooth CW complexes
In this paper we present the notion of smooth CW complexes given by attaching cubes on the category of diffeological spaces, and we study their smooth homotopy structures related to the homotopy
Model categories of smooth spaces
We construct a model structure on the category of diffeological spaces whose weak equivalences are just smooth maps inducing isomorphisms on smooth homotopy groups. Our approach applies to the
Amodel for the ∞-category of stratified spaces
Let P be a poset. In this note we define a combinatorial simplicial model structure on the category of simplicial sets over the nerve of P whose underlying ∞category is the∞-category of P-stratified
Local systems in diffeology
By using local systems over simplicial sets with values in differential graded algebras, we consider a framework of rational and R-local homotopy theory for diffeological spaces with arbitrary
A model structure on the category of diffeological spaces
We construct a model category structure on the category of diffeological spaces which is Quillen equivalent to the model structure on the category of topological spaces based on the notions of Serre
Long exact sequences for de Rham cohomology of diffeological spaces
In this paper we present the notion of de Rham cohomology with compact support for diffeological spaces. Moreover we shall discuss the existence of three long exact sequences. As a concrete example,
On the category of stratifolds
Stratifolds are considered from a categorical point of view. We show among others that the category of stratifolds fully faithfully embeds into the category of ${\mathbb R}$-algebras as does the
An Introduction to Diffeology
This text presents the basics of Diffeology and themain domains: Homotopy, FiberBundles,Quotients, Singularities, Cartan-deRhamCalculus —which form the core of differential geometry— from the point
A model for the $\infty$-category of stratified spaces
Let $P$ be a poset. In this note we define a combinatorial simplicial model structure on the category of simplicial sets over the nerve of $P$ whose underlying $\infty$-category is the
...
1
2
3
...

References

SHOWING 1-7 OF 7 REFERENCES
CONFIGURATION SPACES WITH PARTIALLY SUMMABLE LABELS AND HOMOLOGY THEORIES
It is shown that any subset of a topological abelian monoid gives rise to a generalized homology theory that is closely related to the notion of labeled configuration space. Applications of the main
STEENROD-ČECH HOMOLOGY-COHOMOLOGY THEORIES ASSOCIATED WITH BIVARIANT FUNCTORS
Let NG0 denote the category of all pointed numerically generated spaces and continuous maps preserving base-points. In [SYH], we described a passage from bivariant functors NG 0 ×NG0 → NG0 to
Equivariant Stable Homotopy Theory
The last decade has seen a great deal of activity in this area. The chapter provides a brief sketch of the basic concepts of space-level equivariant homotopy theory. It also provides an introduction
Convenient Categories of Smooth Spaces
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold,
QUASIFASERUNGEN UND UNENDLICHE SYMMETRISCHE PRODUKTE