scholarly journals Homotopy theory of Moore flows (II)

2021 ◽  
Vol 36 (2) ◽  
pp. 157-239
Author(s):  
Philippe Gaucher
Keyword(s):  
Internal Structure ◽  
Homotopy Theory ◽  
Left Adjoint ◽  
Main Step ◽  
Derived Functor ◽  
Category Equivalence ◽  
Q Model ◽  
Quillen Equivalent ◽  
Model Structures

This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.

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 of cyclic presheaves

2010 ◽  
Vol 107 (1) ◽  
pp. 30 ◽  
Author(s):  
Sigurd Seteklev ◽  
Paul Arne Østvær
Keyword(s):  
Homotopy Theory ◽  
Local Model ◽  
Simplicial Presheaves ◽  
Model Structures ◽  
Cyclic Sets

We generalize the homotopy theory of cyclic sets to cyclic presheaves on small Grothendieck sites. This is achieved by constructing pointwise and local model structures reminiscent of the homotopy theory of simplicial presheaves.

scholarly journals Quillen model structures for relative homological algebra

2002 ◽  
Vol 133 (2) ◽  
pp. 261-293 ◽  
Author(s):  
J. DANIEL CHRISTENSEN ◽  
MARK HOVEY
Keyword(s):  
Homotopy Theory ◽  
Homological Algebra ◽  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Projective Class ◽  
Chain Complexes ◽  
Model Structures

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

scholarly journals On the structure of simplicial categories associated to quasi-categories

2011 ◽  
Vol 150 (3) ◽  
pp. 489-504 ◽  
Author(s):  
EMILY RIEHL
Keyword(s):  
Left Adjoint ◽  
3 Dimensional ◽  
Simplicial Sets ◽  
Simplicial Set ◽  
Simplicial Category ◽  
Model Structures

AbstractThe homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

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.

Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

2004 ◽  
Vol 12 (3) ◽  
pp. 225-243 ◽  
Author(s):  
J. M. Garcia-Calcines ◽  
M. Garcia-Pinillos ◽  
L. J. Hernandez-Paricio
Keyword(s):  
Homotopy Theory ◽  
Proper Homotopy ◽  
Model Structures

X-ray microtomography

1988 ◽  
Vol 46 ◽  
pp. 998-999
Author(s):  
H.W. Deckman ◽  
B.F. Flannery ◽  
J.H. Dunsmuir ◽  
K.D' Amico
Keyword(s):  
Computed Tomography ◽  
Internal Structure ◽  
Imaging Technique ◽  
Three Dimensional ◽  
Tissue Structure ◽  
Individual Element ◽  
X Ray ◽  
Non Invasive ◽  
Panoramic Imaging ◽  
Three Dimensional Images

We have developed a new X-ray microscope which produces complete three dimensional images of samples. The microscope operates by performing X-ray tomography with unprecedented resolution. Tomography is a non-invasive imaging technique that creates maps of the internal structure of samples from measurement of the attenuation of penetrating radiation. As conventionally practiced in medical Computed Tomography (CT), radiologists produce maps of bone and tissue structure in several planar sections that reveal features with 1mm resolution and 1% contrast. Microtomography extends the capability of CT in several ways. First, the resolution which approaches one micron, is one thousand times higher than that of the medical CT. Second, our approach acquires and analyses the data in a panoramic imaging format that directly produces three-dimensional maps in a series of contiguous stacked planes. Typical maps available today consist of three hundred planar sections each containing 512x512 pixels. Finally, and perhaps of most import scientifically, microtomography using a synchrotron X-ray source, allows us to generate maps of individual element.

The internal structure of medullated wool

1984 ◽  
Vol 42 ◽  
pp. 388-389
Author(s):  
Leo Barish
Keyword(s):  
Light Microscopy ◽  
Internal Structure ◽  
Wool Fiber ◽  
Guard Hair ◽  
Bright Field ◽  
Wool Fibers ◽  
Thin Skin ◽  
Air Pockets

Although most of the wool used today consists of fine, unmedullated down-type fibers, a great deal of coarse wool is used for carpets, tweeds, industrial fabrics, etc. Besides the obvious diameter difference, coarse wool fibers are often medullated.Medullation may be easily observed using bright field light microscopy. Fig. 1A shows a typical fine diameter nonmedullated wool fiber, Fig. IB illustrates a coarse fiber with a large medulla. The opacity of the medulla is due to the inability of the mounting media to penetrate to the center of the fiber leaving air pockets. Fig. 1C shows an even thicker fiber with a very large medulla and with very thin skin. This type of wool is called “Kemp”, is shed annually or more often, and corresponds to guard hair in fur-bearing animals.

Localization of DNA in chromatin using ESI and osmium ammine staining

1990 ◽  
Vol 48 (3) ◽  
pp. 116-117
Author(s):  
C.L. Woodcock ◽  
R.A. Horowitz ◽  
D. P. Bazett-Jones ◽  
A.L. Olins
Keyword(s):  
Spectroscopic Imaging ◽  
Energy Losses ◽  
Electron Spectroscopic Imaging ◽  
Feulgen Reaction ◽  
Specific Location ◽  
Eukaryotic Nucleus ◽  
Chromatin Fibers ◽  
Patiria Miniata ◽  
Model Structures

In the eukaryotic nucleus, DNA is packaged into nucleosomes, and the nucleosome chain folded into ‘30nm’ chromatin fibers. A number of different model structures, each with a specific location of nucleosomal and linker DNA have been proposed for the arrangment of nucleosomes within the fiber. We are exploring two strategies for testing the models by localizing DNA within chromatin: electron spectroscopic imaging (ESI) of phosphorus atoms, and osmium ammine (OSAM) staining, a method based on the DNA-specific Feulgen reaction.Sperm were obtained from Patiria miniata (starfish), fixed in 2% GA in 150mM NaCl, 15mM HEPES pH 8.0, and embedded In Lowiciyl K11M at -55C. For OSAM staining, sections 100nm to 150nm thick were treated as described, and stereo pairs recorded at 40,000x and 100KV using a Philips CM10 TEM. (The new osmium ammine-B stain is available from Polysciences Inc). Uranyl-lead (U-Pb) staining was as described. ESI was carried out on unstained, very thin (<30 nm) beveled sections at 80KV using a Zeiss EM902. Images were recorded at 20,000x and 30,000x with median energy losses of 110eV, 120eV and 160eV, and a window of 20eV.

Categorical Homotopy Theory

2009 ◽  
Author(s):  
Emily Riehl
Keyword(s):  
Homotopy Theory
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