| Title:
|
Topos based homology theory (English) |
| Author:
|
Mielke, M. V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
3 |
| Year:
|
1993 |
| Pages:
|
549-565 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
singular homology |
| Keyword:
|
homotopy |
| Keyword:
|
excision |
| Keyword:
|
topos |
| Keyword:
|
interval |
| MSC:
|
18G99 |
| MSC:
|
55N10 |
| MSC:
|
55N35 |
| MSC:
|
55N40 |
| MSC:
|
55U40 |
| idZBL:
|
Zbl 0785.55003 |
| idMR:
|
MR1243087 |
| . |
| Date available:
|
2009-01-08T18:06:08Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118612 |
| . |
| Reference:
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