# Weak complicial sets I. Basic homotopy theory

@article{Verity2008WeakCS, title={Weak complicial sets I. Basic homotopy theory}, author={Dominic R. Verity}, journal={Advances in Mathematics}, year={2008}, volume={219}, pages={1081-1149} }

This paper develops the foundations of a simplicial theory of weak ?-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ?-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict ?-categories to provide an armoury of well… Expand

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#### References

SHOWING 1-10 OF 28 REFERENCES

Weak Complicial Sets and Internal Quasi-Categories

- 2007

It is well known that we may represent (strict) ω-categories as simplicial sets, via Street’s ω-categorical nerve construction [2]. What may be less well known, is that we may extend Street’s nerve… Expand

Weak omega-categories

- 2003

This paper proposes to define a weak higher-dimensional category to be a simplicial set satisfying properties. The definition is a refinement of that suggested at the end of [St3] which required… Expand

CHAPTER 2 – Homotopy Theories and Model Categories

- Mathematics
- 1995

This chapter explains homotopy theories and model categories. A model category is just an ordinary category with three specified classes of morphisms—fibrations, cofibrations, and weak… Expand

Vogt's theorem on categories of homotopy coherent diagrams

- 2007

Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the… Expand

BASIC CONCEPTS OF ENRICHED CATEGORY THEORY

- Philosophy
- 2005

Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory,… Expand

Quasi-categories and Kan complexes

- Mathematics
- 2002

A quasi-category X is a simplicial set satisfying the restricted Kan conditions of Boardman and Vogt. It has an associated homotopy category hoX. We show that X is a Kan complex iff hoX is a… Expand

Handbook of algebraic topology

- Mathematics
- 1995

Foreword. List of Contributors. Homotopy types (H.-J. Baues). Homotopy theories and model categories (W.G. Dwyer, J. Spalinski). Proper homotopy theory (T. Porter). Introduction to fibrewise homotopy… Expand

On injectivity in locally presentable categories

- Mathematics
- 1993

AbstractWe show that some fundamental results about projectivity classes, weakly coreﬂective subcate-gories and cotorsion theories can be generalized from R -modules to arbitrary locally… Expand

Accessible categories : the foundations of categorical model theory

- Mathematics
- 1989

[F-S] D. Fremlin and S. Shelah, Pointwise compact and stable sets of measurable functions, manuscript, 1990. [G-G-M-S] N. Ghoussoub, G. Godefroy, B. Maurey, W. Schachermayer, Some topological and… Expand

The algebra of oriented simplexes

- Mathematics
- 1987

Abstract Anm-simplex x in ann-category A consists of the assignment of anr-cell x(u) to each (r + 1)-element subset u of {0, 1,..., m} such that the source and target (r−1)-cells of x(u) are… Expand