Homotopy Theory of Simplicial Sets

2015 ◽  
pp. 15-27
Author(s):  
John F. Jardine
Keyword(s):  
Homotopy Theory ◽  
Simplicial Sets

Algebraic Homotopy Theory

1981 ◽  
Vol 33 (2) ◽  
pp. 302-319 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Homotopy Type ◽  
Homotopy Theory ◽  
Unique Factorization ◽  
Contravariant Functor ◽  
Unique Factorization Domain ◽  
Simplicial Sets ◽  
Simplicial Set ◽  
Image Position ◽  
Algebraic Homotopy Theory

Kan and Miller have shown in [9] that the homotopy type of a finite simplicial set K can be recovered from its R-algebra of 0-forms A0K, when R is a unique factorization domain. More precisely, if is the category of simplicial sets and is the category of R-algebras there is a contravariant functorwiththe simplicial set homomorphisms from X to the simplicial R-algebra ∇, whereand the faces and degeneracies of ∇ are induced byandrespectively.

scholarly journals A strictly commutative model for the cochain algebra of a space

2020 ◽  
Vol 156 (8) ◽  
pp. 1718-1743
Author(s):  
Birgit Richter ◽  
Steffen Sagave
Keyword(s):  
Homotopy Theory ◽  
Rational Homotopy ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Simplicial Sets ◽  
Differential Graded Algebras ◽  
Chain Complexes ◽  
Simplicial Set ◽  
New Construction ◽  
Integral Version

AbstractThe commutative differential graded algebra $A_{\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal {I}}(X)$ of $A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal {I}$ to model $E_{\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\mathcal {I}$-dgas. We define a functor $A^{\mathcal {I}}$ from simplicial sets to commutative $\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty }$ dga of cochains. The functor $A^{\mathcal {I}}$ shares many properties of $A_{\mathrm {PL}}$, and can be viewed as a generalization of $A_{\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

scholarly journals Dwyer–Kan homotopy theory for cyclic operads

2021 ◽  
pp. 1-30
Author(s):  
Gabriel C. Drummond-Cole ◽  
Philip Hackney
Keyword(s):  
Homotopy Theory ◽  
Forgetful Functor ◽  
General Definition ◽  
Model Structure ◽  
Simplicial Sets ◽  
Explicit Formulae

Abstract We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.

scholarly journals Homotopy theory of complete Lie algebras and Lie models of simplicial sets

2018 ◽  
Vol 11 (3) ◽  
pp. 799-825
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Aniceto Murillo ◽  
Daniel Tanré
Keyword(s):  
Lie Algebras ◽  
Homotopy Theory ◽  
Simplicial Sets

scholarly journals Univalence for inverse diagrams and homotopy canonicity

2014 ◽  
Vol 25 (5) ◽  
pp. 1203-1277 ◽  
Author(s):  
MICHAEL SHULMAN
Keyword(s):  
Natural Number ◽  
Type Theory ◽  
Homotopy Theory ◽  
Partial Answer ◽  
Syntactic Category ◽  
Simplicial Sets ◽  
New Models ◽  
Universe Of Sets

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky's univalent model in simplicial sets, we obtain new models of univalence in a number of (∞, 1)-toposes; this answers a question raised at the Oberwolfach workshop on homotopical type theory. Second, by gluing the syntactic category of univalent type theory along its global sections functor to groupoids, we obtain a partial answer to Voevodsky's homotopy-canonicity conjecture: in 1-truncated type theory with one univalent universe of sets, any closed term of natural number type is homotopic to a numeral.

scholarly journals Semantics of higher inductive types

2019 ◽  
Vol 169 (1) ◽  
pp. 159-208 ◽  
Author(s):  
PETER LEFANU LUMSDAINE ◽  
MICHAEL SHULMAN
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Model Category ◽  
Suitable Model ◽  
Simplicial Sets ◽  
James Construction ◽  
Quillen Model Category ◽  
Wide Range ◽  
Inductive Types ◽  
Cartesian Closed

AbstractHigher inductive typesare a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.We introduce the notion ofcell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category hasweakly stable typal initial algebrasfor any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

scholarly journals Morita homotopy theory for (∞,1)-categories and ∞-operads

2019 ◽  
Vol 31 (3) ◽  
pp. 661-684 ◽  
Author(s):  
Giovanni Caviglia ◽  
Javier J. Gutiérrez
Keyword(s):  
Homotopy Theory ◽  
Model Structure ◽  
Simplicial Sets ◽  
Simplicial Presheaves ◽  
Model Structures ◽  
Simplicial Algebras ◽  
Bousfield Localization

Abstract We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of {(\infty,1)} -categories and {\infty} -operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.

scholarly journals The extended rational homotopy theory of operads

2018 ◽  
Vol 25 (4) ◽  
pp. 493-512
Author(s):  
Benoit Fresse
Keyword(s):  
Homotopy Theory ◽  
American Mathematical Society ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Graded Algebras ◽  
Simplicial Sets ◽  
Differential Graded Algebras ◽  
Rational Cohomology ◽  
Rational Homotopy Type ◽  
Set Up

Abstract In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book [B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups. Part 2. The Applications of (Rational) Homotopy Theory Methods, Math. Surveys Monogr. 217, American Mathematical Society, Providence, 2017]. In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite-type rational cohomology groups).

Stable Homotopy Theory of Simplicial Presheaves

1987 ◽  
Vol 39 (3) ◽  
pp. 733-747 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Closed Model ◽  
Stable Homotopy Theory ◽  
Simplicial Sets ◽  
Stable Homotopy Category ◽  
Closed Model Category ◽  
Simplicial Presheaves ◽  
Direct Use

Let C be an arbitrary Grothendieck site. The purpose of this note is to show that, with the closed model structure on the category S Pre(C) of simplicial presheaves in hand, it is a relatively simple matter to show that the category S Pre(C)stab of presheaves of spectra (of simplicial sets) satisfies the axioms for a closed model category, giving rise to a stable homotopy theory for simplicial presheaves. The proof is modelled on the corresponding result for simplicial sets which is given in [1], and makes direct use of their Theorem A.7.This result gives a precise description of the associated stable homotopy category Ho(S Pre(C))stab, according to well known results of Quillen [6]. One will recall, however, that it is preferable to have several different descriptions of the stable homotopy category, for the construction of smash products and the like.

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