Weak complicial sets I. Basic homotopy theory

@article{Verity2008WeakCS,
title={Weak complicial sets I. Basic homotopy theory},
author={Dominic R. Verity},
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

Figures from this paper

Weak complicial sets, a simplicial weak omega-category theory. Part II: nerves of complicial Gray-categories
This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generaliseExpand
A pr 2 00 6 Weak Complicial Sets A Simplicial Weak ω-Category Theory Part II : Nerves of Complicial Gray-Categories
This paper continues the development of a simplicial theory of weak ω-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise bothExpand
Complicial Sets, an Overture
The aim of these notes is to introduce the intuition motivating the notion of a complicial set, a simplicial set with certain marked “thin” simplices that witness a composition relation between theExpand
Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves
• Mathematics, Computer Science
• Appl. Categorical Struct.
• 2020
It is shown that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit cones, whose shape is governed by the weight for pseudo limits over aHomotopy coherent diagram but with the defining universal property up to equivalence, rather than isomorphism, of mapping spaces. Expand
On the construction of limits and colimits in ∞-categories
• Mathematics
• 2020
In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteriaExpand
Diagrammatic sets and rewriting in weak higher categories.
We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial pasting diagrams, now as a framework for higher-dimensional rewriting and the basis of a model of weak omega-categories.Expand
On the construction of limits and colimits in $\infty$-categories
• Mathematics
• 2018
In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteriaExpand
Fibrations and Yoneda's lemma in an ∞-cosmos
• Mathematics
• 2017
Abstract We use the terms ∞-categories and ∞-functors to mean the objects and morphisms in an ∞-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched categoryExpand
On loop spaces with marking
Here, the loop space functor is defined to be the right adjoint of reduced suspension functor, namely it is a straightforward generalization of the loop space functor of simplicial sets. We willExpand
Weak model categories in classical and constructive mathematics
We introduce a notion of "weak model category". It is a weakening of the classical notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunction, QuillenExpand

References

SHOWING 1-10 OF 28 REFERENCES
Weak Complicial Sets and Internal Quasi-Categories
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 nerveExpand
Weak omega-categories
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 requiredExpand
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 weakExpand
Vogt's theorem on categories of homotopy coherent diagrams
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 theExpand
BASIC CONCEPTS OF ENRICHED CATEGORY THEORY
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
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 aExpand
Handbook of algebraic topology
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 homotopyExpand
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 locallyExpand
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 andExpand
The algebra of oriented simplexes
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) areExpand