Bivariant cyclic cohomology and models for cyclic homology types

AbstractThis paper concerns types of algebraic objects, such as mixed complexes and S-modules, which are used to obtain the homology and cohomology of interest in cyclic homology theory. We prove that the following five categories are equivalent: The derived category of mixed complexes. The homotopy category of free mixed complexes. The derived category of S-modules. The homotopy category of divisible S-modules. The homotopy category of special towers of supercomplexes. Thus any of these categories represents the category of cyclic homotopy types

Postnikov extensions of ring spectra

We give a functorial construction of k-invariants for ring spectra and use these to classify extensions in the Postnikov tower of a ring spectrum.Comment: This is the version published by Algebraic & Geometric Topology on 1 November 200

A curious example of two model categories and some associated differential graded algebras

The paper gives a new proof that the model categories of stable modules for the rings Z/(p^2) and (Z/p)[\epsilon]/(\epsilon^2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K-theories

Rigidification of quasi-categories

We give a new construction for rigidifying a quasi-category into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an application of these methods, we use our model to reprove some basic facts from Lurie's "Higher Topos Theory" regarding the rigidification process.Comment: 26 page

Transmission of a Seismic Wave generated by impacts on Granular Asteroids

In this paper we use a Soft-Sphere Discrete Element method code to simulate the transmission and study the attenuation of a seismic wave. Then, we apply our findings to the different space missions that have had to touch the surface of different small bodies. Additionally, we do the same in regards to the seismic wave generated by the hypervelocity impacts produced by the DART and Hayabusa2 missions once the shock wave transforms into a seismic wave. We find that even at very low pressures, such as those present in the interior of asteroids, the seismic wave speed can still be on the order of hundreds of m/s depending on the velocity of the impact that produces the wave. As expected from experimental measurements, our results show that wave velocity is directly dependent on $P^{1/6}$, where $P$ is the total pressure (confining pressure plus wave induced pressure). Regardless of the pressure of the system and the velocity of the impact (in the investigated range), energy dissipation is extremely high. These results provide us with a way to anticipate the extent to which a seismic wave could have been capable of moving some small particles on the surface of a small body upon contact with a spacecraft. Additionally, this rapid energy dissipation would imply that even hypervelocity impacts should perturb only the external layer of a self-gravitating aggregate on which segregation and other phenomena could take place. This would in turn produce a layered structure of which some evidence has been observedComment: Accepted for publication in The Planetary Sciences Journa

Duality and Pro-Spectra

Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier-Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-34.abs.htm

Obstruction Theory in Model Categories

Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.Comment: 17 pages. v3 includes improved introduction and several other minor improvement

Discovery of a 500 pc shell in the nucleus of Centaurus A

Spitzer Space Telescope mid-infrared images of the radio galaxy Centaurus A reveal a shell-like, bipolar, structure 500 pc to the north and south of the nucleus. This shell is seen in 5.8, 8.0 and 24 micron broad-band images. Such a remarkable shell has not been previously detected in a radio galaxy and is the first extragalactic nuclear shell detected at mid-infrared wavelengths. We estimate that the shell is a few million years old and has a mass of order million solar masses. A conservative estimate for the mechanical energy in the wind driven bubble is 10^53 erg. The shell could have created by a small few thousand solar mass nuclear burst of star formation. Alternatively, the bolometric luminosity of the active nucleus is sufficiently large that it could power the shell. Constraints on the shell's velocity are lacking. However, if the shell is moving at 1000 km/s then the required mechanical energy would be 100 times larger.Comment: submitted to ApJ Letter

On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes

We show that if $G$ is an infinitesimal elementary supergroup scheme of height $\leq r$, then the cohomological spectrum $|G|$ of $G$ is naturally homeomorphic to the variety $\mathcal{N}_r(G)$ of supergroup homomorphisms $\rho: \mathbb{M}_r \rightarrow G$ from a certain (non-algebraic) affine supergroup scheme $\mathbb{M}_r$ into $G$. In the case $r=1$, we further identify the cohomological support variety of a finite-dimensional $G$-supermodule $M$ as a subset of $\mathcal{N}_1(G)$. We then discuss how our methods, when combined with recently-announced results by Benson, Iyengar, Krause, and Pevtsova, can be applied to extend the homeomorphism $\mathcal{N}_r(G) \cong |G|$ to arbitrary infinitesimal unipotent supergroup schemes.Comment: Fixed some algebra misidentifications, primarily in Sections 1.3 and 3.3. Simplified the proof of Proposition 3.3.

The Warped Disk of Centaurus A from a Radius of 2 to 6500pc

We compile position and inclination angles for tilted ring fits to the warped dusty and gaseous disk of Cen A, spanning a radius of 1.8 to 6500 pc, from recent observations. For radii exterior to 1 kpc, tilted-ring orientations lie on an arc, on a plot of polar-inclination versus position-angle, suggesting that precession following a merger can account for the ring morphology. Three kinks in the ring orientations are seen on the polar plot, the one at radius of about 1.3 kpc we suspect corresponds to the location where self-gravity in the disk affects the ring precession rate. Another at a radius of about 600 pc may be associated with a gap in the gas distribution. A third kink is seen at a radius of 100 pc. A constant inclination tilted disk precessing about the jet axis may describe the disk between 100 and 20 pc but not interior to this. A model with disk orientation matching the molecular circumnuclear disk at 100 pc that decays at smaller radii to an inner flat disk perpendicular to the jet may account for disk orientations within 100 pc. Neither model would account for the cusps or changes in disk orientation at 100 or 600 pc