A spectral sequence for spaces of maps between operads

We construct a tower of fibrations approximating the derived mapping space between two simplicially enriched operads subject to mild conditions. The n-th stage of the tower is obtained by neglecting operations with more than n inputs. The main theorem describes the layers of this tower.Comment: v2: some typos corrected, some simplifications, bibliography improve

Chromatic homotopy theory is asymptotically algebraic

Inspired by the Ax--Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the $E(n,p)$-local categories over any non-prinicipal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.Comment: Minor changes, to appear in Inventiones Mathematica

Fibers of partial totalizations of a pointed cosimplicial space

Let $X^\bullet$ be a cosimplicial object in a pointed $\infty$-category. We show that the fiber of $\mathrm{Tot}_m(X^\bullet) \to \mathrm{Tot}_n(X^\bullet)$ depends only on the pointed cosimplicial object $\Omega^k X^\bullet$ and is in particular a $k$-fold loop object, where $k = 2n - m+2$. The approach is explicit obstruction theory with quasicategories. We also discuss generalizations to other types of homotopy limits and colimits.Comment: 14 pages. Final version, to appear in Proc. Amer. Math. So

Suspicious minds : the dramatisation of paranoia in Victorian poetry : a thesis presented in partial fulfilment of the requirements for the degree of Master of Arts in English at Massey University

This thesis contains readings of a number of Victorian poems by Alfred Tennyson, Robert Browning and Dante Gabriel Rossetti which dramatise paranoia and jealousy. A range of twentieth-century theories of paranoia (including clinical, Freudian and Lacanian) have been used as explanatory tools for interpreting the representations of paranoia in the poems. The reading of Tennyson's Maud is based on Freud's theory of homoerotic motives. The reading of Browning's "'Childe Roland to the Dark Tower Came'" is based on the Lacanian concepts of foreclosure and the Name-of-the-Father. The readings of the jealousy poems are based on both theories, and this section includes a discussion of the limitations of the theories as explanatory tools. The general approach has been to apply clinical and psychoanalytical constructs and explanations to each poem separately, although there is some discussion involving the comparison of paranoid behaviours and motives across all the poems. Areas for further research are suggested in the concluding chapter

On local-to-global spectral sequences for the cohomology of diagrams

The aim of this paper is to construct and examine three candidates for local-to-global spectral sequences for the cohomology of diagrams of algebras with directed indexing. In each case, the $E^2$ -terms can be viewed as a type of local cohomology relative to a map or an object in the diagram.Comment: 27 pages; incorporated corrections based on referee's report and added a few reference

Quantale-valued Cauchy tower spaces and completeness

[EN] Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.Jäger, G.; Ahsanullah, TMG. (2021). Quantale-valued Cauchy tower spaces and completeness. Applied General Topology. 22(2):461-481. https://doi.org/10.4995/agt.2021.15610OJS461481222J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.T. M. G. Ahsanullah and G. Jäger, Probabilistic uniform convergence spaces redefined, Acta Math. 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