Homotopy Coherent Category Theory

Homotopy Coherent Category Theory (1996)

Cached

Download Links

[www.bangor.ac.uk]

by Jean-marc Cordier , Timothy Porter , Wales U. K

Citations:

19 - 6 self

Versions

Version History

Metadata Version 0

Datum	Value	Source
TITLE	33 rue Saint Leu,	SVM HeaderParse 0.1
AUTHOR NAME	Jean-marc Cordier	SVM HeaderParse 0.1
AUTHOR NAME	Timothy Porter	SVM HeaderParse 0.1
AUTHOR ADDR	; Dean Street, BANGOR,; Gwynedd, LL57 1UT,	SVM HeaderParse 0.1
AUTHOR NAME	Wales U. K	SVM HeaderParse 0.1
ABSTRACT	In homotopy theory, one often needs to "do " universal algebra "up to homotopy", for instance in the theory of homotopy everything H-spaces. Universal algebra nowadays is most easily expressed in categorical terms, so this calls for a form of category theory "up to homotopy", (cf. Heller [34]). In studying the homotopy theory of compact metric spaces, there is the classically known complication that the assignment of the nerve of an open cover to a cover is only a functor up to homotopy. Thus in shape theory, [39], one does not have a very rich underlying homotopy theory. Strong shape (cf. Lisica and Mardesi'c, [37]) and Steenrod homotopy (cf. Edwards and Hastings [27]) have a richer theory but at the cost of much harder proofs. Shape theory has been interpreted categorically in an elegant way. This provides an overview of most aspects of the theory, as well as the basic proobject formulation that it shares with 'etale homotopy and the theory of derived categories. Can one perform a similar process with strong shape and hence provide tools for enriching that theory, rigidifying 'etale 1 homotopy and enriching derived categories? We note, in particular, Grothendieck's plan for the theory of derived categories, sketched out in Pursuing Stacks, [31], which bears an uncanny ressemblance to the view of homotopy theory put forward by Heller in [34]. Grothendieck's `Pursuing Stacks ' program, in fact, again raises the spectre of doing categorical construction `up to homotopy', as his image of a stack is as a sheaf `up to homotopy ' in which the stalks are algebraic models of homotopy types and the whole object has geometric meaning. Problems of homotopy coherence also arise naturally in studying equivariant homotopy theory. If G is a group then the equivariant homotopy of G-complexes can be studied in	SVM HeaderParse 0.1
CITATIONS	48 found	ParsCit 1.0