scholarly journals Dendroidal spaces, Γ-spaces and the special Barratt--Priddy--Quillen theorem

2020 ◽  
Vol 2020 (760) ◽  
pp. 229-265 ◽  
Author(s):  
Pedro Boavida de Brito ◽  
Ieke Moerdijk
Keyword(s):  
Homotopy Theory ◽  
Model Structure ◽  
Covariant Model

AbstractWe study the covariant model structure on dendroidal spaces, and establish direct relations to the homotopy theory of algebras over a simplicial operad as well as to the homotopy theory of special Γ-spaces. As an important tool in the latter comparison, we present a sharpening of the classical Barratt–Priddy–Quillen theorem.

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 well-generated algebraic triangulated categories

2008 ◽  
Vol 3 (1) ◽  
pp. 53-75 ◽  
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Regular Cardinal ◽  
Homotopy Theory ◽  
Model Category ◽  
Triangulated Categories ◽  
Model Structure ◽  
Quillen Model Category ◽  
Quillen Model

AbstractFor every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially dg categories which are stable under suspensions, cosuspensions, cones and α-small sums.Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.

scholarly journals Arrow categories of monoidal model categories

2019 ◽  
Vol 125 (2) ◽  
pp. 185-198
Author(s):  
David White ◽  
Donald Yau
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Topological Spaces ◽  
Model Structure ◽  
Model Categories ◽  
Projective Model ◽  
Monoidal Structure ◽  
Chain Complexes

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, in the course of his work on Smith ideals. As a corollary, we prove that the projective model structure in cubical homotopy theory is a monoidal model structure. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, pro-categories, and topological spaces.

scholarly journals Model structure on the universe of all types in interval type theory

2020 ◽  
pp. 1-32
Author(s):  
Simon Boulier ◽  
Nicolas Tabareau
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Model Structure ◽  
Model Categories ◽  
Model Interpretation ◽  
Slight Generalization ◽  
Cubical Sets ◽  
Definition Of ◽  
Interval Type ◽  
The Universe

Abstract Model categories constitute the major context for doing homotopy theory. More recently, homotopy type theory (HoTT) has been introduced as a context for doing syntactic homotopy theory. In this paper, we show that a slight generalization of HoTT, called interval type theory (⫿TT), allows to define a model structure on the universe of all types, which, through the model interpretation, corresponds to defining a model structure on the category of cubical sets. This work generalizes previous works of Gambino, Garner, and Lumsdaine from the universe of fibrant types to the universe of all types. Our definition of ⫿TT comes from the work of Orton and Pitts to define a syntactic approximation of the internal language of the category of cubical sets. In this paper, we extend the work of Orton and Pitts by introducing the notion of degenerate fibrancy, which allows to define a fibrant replacement, at the heart of the model structure on the universe of all types. All our definitions and propositions have been formalized using the Coq proof assistant.

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 UNITARY FUNCTOR CALCULUS WITH REALITY

2021 ◽  
pp. 1-34
Author(s):  
NIALL TAGGART
Keyword(s):  
Homotopy Theory ◽  
Model Structure ◽  
Classifying Space ◽  
The Real ◽  
Calculus Of Functors ◽  
Taylor Tower ◽  
Orthogonal Calculus

Abstract We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study ‘functors with reality’ such as the Real classifying space functor, $\BU_\Bbb{R}(-)$ . The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of $C_2 \ltimes \U(n)$ . We further give model categorical considerations, producing a zigzag of Quillen equivalences between spectra with an action of $C_2 \ltimes \U(n)$ and a model structure on the category of input functors which captures the homotopy theory of the n-th layer of the Taylor tower.

scholarly journals Homotopy theory of Moore flows (I)

2021 ◽  
Vol 3 ◽  
pp. 3
Author(s):  
Philippe Gaucher
Keyword(s):  
Tensor Product ◽  
Commutative Semigroup ◽  
Homotopy Theory ◽  
Model Category ◽  
Model Structure ◽  
Q Model ◽  
Quillen Equivalent

A reparametrization category is a small topologically enriched symmetric semimonoidal category such that the semimonoidal structure induces a structure of a commutative semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the closed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed d-spaces and the q-model structure of flows.

scholarly journals Homotopy theory for truncated weak equivalences of simplicial groups

1997 ◽  
Vol 121 (1) ◽  
pp. 51-74 ◽  
Author(s):  
ANTONIO R. GARZÓN ◽  
JESÚS G. MIRANDA
Keyword(s):  
Homotopy Theory ◽  
Model Structure ◽  
Simplicial Homotopy

In this paper we give for any r, n, 0 [les ] r [les ] n, a Quillen's model structure to the category of simplicial groups where the weak equivalences are those morphisms f[bull ] such that πq(f[bull ]) is an isomorphism for r [les ] q [les ] n. This is carried out by studying the cases r = 0 and n → ∞ previously and, in each one of them, we make explicit some constructions for the associated homotopy theories, such as the cylinder and path objects and the loop and suspension functors, and we also relate the simplicial homotopy relation to the homotopy relation obtained from these structures.

scholarly journals Dendroidal sets as models for connective spectra

2014 ◽  
Vol 14 (3) ◽  
pp. 387-421 ◽  
Author(s):  
Matija Bašić ◽  
Thomas Nikolaus
Keyword(s):  
Homotopy Theory ◽  
Model Structure ◽  
Combinatorial Model

AbstractDendroidal sets have been introduced as a combinatorial model for homotopy coherent operads. We introduce the notion of fully Kan dendroidal sets and show that there is a model structure on the category of dendroidal sets with fibrant objects given by fully Kan dendroidal sets. Moreover we show that the resulting homotopy theory is equivalent to the homotopy theory of connective spectra.

Lattice imaging and pore structure of β ferric oxyhydroxide

1978 ◽  
Vol 36 (1) ◽  
pp. 264-265
Author(s):  
T. Baird ◽  
J.R. Fryer ◽  
S.T. Galbraith
Keyword(s):  
Electron Microscopy ◽  
Room Temperature ◽  
Bulk Material ◽  
Model Structure ◽  
Bulk Crystals ◽  
Lattice Imaging ◽  
Thin Sections ◽  
Ferric Oxyhydroxide ◽  
Resolution Electron ◽  
Hydrolysis Of

Introduction Previously we had suggested (l) that the striations observed in the pod shaped crystals of β FeOOH were an artefact of imaging in the electron microscope. Contrary to this adsorption measurements on bulk material had indicated the presence of some porosity and Gallagher (2) had proposed a model structure - based on the hollandite structure - showing the hollandite rods forming the sides of 30Å pores running the length of the crystal. Low resolution electron microscopy by Watson (3) on sectioned crystals embedded in methylmethacrylate had tended to support the existence of such pores.We have applied modern high resolution techniques to the bulk crystals and thin sections of them without confirming these earlier postulatesExperimental β FeOOH was prepared by room temperature hydrolysis of 0.01M solutions of FeCl3.6H2O, The precipitate was washed, dried in air, and embedded in Scandiplast resin. The sections were out on an LKB III Ultramicrotome to a thickness of about 500Å.

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