Homotopy theory of posets

@article{Raptis2010HomotopyTO,
  title={Homotopy theory of posets},
  author={George E. Raptis},
  journal={Homology, Homotopy and Applications},
  year={2010},
  volume={12},
  pages={211-230}
}
  • G. Raptis
  • Published 2010
  • Mathematics
  • Homology, Homotopy and Applications
This paper studies the category of posets Pos as a model for the homotopy theory of spaces. We prove that: (i) Pos admits a (cofibrantly generated and proper) model structure and the inclusion functor Pos ↪→ Cat into Thomason’s model category is a right Quillen equivalence, and (ii) there is a proper class of different choices of cofibrations for a model structure on Pos or Cat where the weak equivalences are defined by the nerve functor. We also discuss the homotopy theory of posets from the… 
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References

SHOWING 1-10 OF 23 REFERENCES
Fibrations of Simplicial Sets
  • Tibor Beke
  • Mathematics
    Appl. Categorical Struct.
  • 2010
TLDR
To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan’s Ex functor act on graphs and on nerves of small categories.
On Homotopy Types of Alexandroff Spaces
TLDR
This work characterize pairs of spaces X,Y such that the compact-open topology on C(X,Y) is Alexandroff, gives a homotopy type classification of a class of infinite Alexandroffer spaces and proves some results concerning cores of locally finite spaces.
Sheafifiable homotopy model categories
  • Tibor Beke
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2000
If a Quillen model category can be specified using a certain logical syntax (intuitively, ‘is algebraic/combinatorial enough’), so that it can be defined in any category of sheaves, then the
Les Pr'efaisceaux comme mod`eles des types d''homotopie
Grothendieck introduced in Pursuing Stacks the notion of test category . These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes. A well
Homotopy inverses for nerve
Whitehead [19] introduced the category of CW complexes as the appropriate category in which to do homotopy theory. Eilenberg, Mac Lane and Zilber ([1], [2]) defined the notion of simplicial set in
Simple homotopy types and finite spaces
Whitehead’s theory of simple homotopy types is inspired by Tietze’s theorem in combinatorial group theory, which states that any finite presentation of a group could be deformed into any other by a
Classifying Spaces and Classifying Topoi
This monograph presents a new, systematic treatment of the relation between classifying topoi and classifying spaces of topological categories. Using a new generalized geometric realization which
On Combinatorial Model Categories
TLDR
Some new results about homotopy equivalences, weak equivalences and cofibrations in combinatorial model categories are contributing to this endeavour by some new resultsabout homotope equivalences.
Simplicial Homotopy Theory
TLDR
Simplicial sets, model categories, and cosimplicial spaces: applications for homotopy coherence, results and constructions, and more.
La classe des morphismes de Dwyer n'est pas stable par rétractes
In [1] R. Thomason claims that a retract of a Dwyer map is a Dwyer map to show the closed model category structure he defined on the category of small categories is proper. This paper gives a
...
...