Topos based homology theory
Mielke, M. V.
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993), p. 549-565 / Harvested from Czech Digital Mathematics Library
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In this paper we extend the Eilenberg-Steenrod axiomatic description of a homology theory from the category of topological spaces to an arbitrary category and, in particular, to a topos. Implicit in this extension is an extension of the notions of homotopy and excision. A general discussion of such homotopy and excision structures on a category is given along with several examples including the interval based homotopies and, for toposes, the excisions represented by ``cutting out'' subobjects. The existence of homology theories on toposes depends upon their internal logic. It is shown, for example, that all ``reasonable'' homology theories on a topos in which De Morgan's law holds are trivial. To obtain examples on non-trivial homology theories we consider singular homology based on a cosimplicial object. For toposes singular homology satisfies all the axioms except, possibly, excision. We introduce a notion of ``tightness'' and show that singular homology based on a sufficiently tight cosimplicial object satisfies the excision axiom. Cha\-rac\-terizations of various types of tight cosimplicial objects in the functor topos $\text{\rm Sets}^C$ are given and, as a result, a general method for constructing non-trivial homology theories is obtained. We conclude with several explicit examples.

Publié le : 1993-01-01
Classification:  18G99,  55N10,  55N35,  55N40,  55U40 
@article{118612,
     author = {M. V. Mielke},
     title = {Topos based homology theory},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {34},
     year = {1993},
     pages = {549-565},
     zbl = {0785.55003},
     mrnumber = {1243087},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118612}
}

Mielke, M. V. Topos based homology theory. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 549-565. http://gdmltest.u-ga.fr/item/118612/

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