2,175 research outputs found

### Stems and Spectral Sequences

We introduce the category Pstem[n] of n-stems, with a functor P[n] from
spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of
a space. We show how to associate to each simplicial n-stem Q an
(n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem
of a simplicial space X, the truncated spectral sequence for Q is the
truncation of the usual homotopy spectral sequence of X. Similar results are
also proven for cosimplicial n-stems. They are helpful for computations, since
n-stems in low degrees have good algebraic models

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### An algorithm for the optimisation of n-forms using symmetric multilinear forms

This paper describes an algorithm for locating stationary points of n-forms. Use is made of the associated n-linear form, the stationary points of which are seen to coincide with those of the n-form. Conditions of convergence are established using the concept of Liapunov stability, and it is seen that the scheme can always be made to converge to the global maximum of the n-form over unit vectors

### Cosimplicial resolutions and homotopy spectral sequences in model categories

We develop a general theory of cosimplicial resolutions, homotopy spectral
sequences, and completions for objects in model categories, extending work of
Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a
generalized cosimplicial version of the Dwyer-Kan-Stover theory of resolution
model categories, and we are able to construct our homotopy spectral sequences
and completions using very flexible weak resolutions in the spirit of relative
homological algebra. We deduce that our completion functors have triple
structures and preserve certain fiber squares up to homotopy. We also deduce
that the Bendersky-Thompson completions over connective ring spectra are
equivalent to Bousfield-Kan completions over solid rings. The present work
allows us to show, in a subsequent paper, that the homotopy spectral sequences
over arbitrary ring spectra have well-behaved composition pairings.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper29.abs.htm

### A chain rule in the calculus of homotopy functors

We formulate and prove a chain rule for the derivative, in the sense of
Goodwillie, of compositions of weak homotopy functors from simplicial sets to
simplicial sets. The derivative spectrum dF(X) of such a functor F at a
simplicial set X can be equipped with a right action by the loop group of its
domain X, and a free left action by the loop group of its codomain Y = F(X).
The derivative spectrum d(E o F)(X)$ of a composite of such functors is then
stably equivalent to the balanced smash product of the derivatives dE(Y) and
dF(X), with respect to the two actions of the loop group of Y. As an
application we provide a non-manifold computation of the derivative of the
functor F(X) = Q(Map(K, X)_+).Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper25.abs.htm

### Group completion and units in I-spaces

The category of I-spaces is the diagram category of spaces indexed by finite
sets and injections. This is a symmetric monoidal category whose commutative
monoids model all E-infinity spaces. Working in the category of I-spaces
enables us to simplify and strengthen previous work on group completion and
units of E-infinity spaces. As an application we clarify the relation to
Gamma-spaces and show how the spectrum of units associated with a commutative
symmetric ring spectrum arises through a chain of Quillen adjunctions.Comment: v3: 43 pages. Minor revisions, accepted for publication in Algebraic
and Geometric Topolog

### The discrete module category for the ring of K-theory operations

We study the category of discrete modules over the ring of degree zero stable
operations in p-local complex K-theory. We show that the p-local K-homology of
any space or spectrum is such a module, and that this category is isomorphic to
a category defined by Bousfield and used in his work on the K-local stable
homotopy category (Amer. J. Math., 1985). We also provide an alternative
characterisation of discrete modules as locally finitely generated modules.Comment: 19 page

### Segal-type algebraic models of n-types

For each n\geq 1 we introduce two new Segal-type models of n-types of
topological spaces: weakly globular n-fold groupoids, and a lax version of
these. We show that any n-type can be represented up to homotopy by such models
via an explicit algebraic fundamental n-fold groupoid functor. We compare these
models to Tamsamani's weak n-groupoids, and extract from them a model for
(k-1)connected n-typesComment: Added index of terminology and notation. Minor amendments and added
details is some definitions and proofs. Some typos correcte

### Complete Boolean algebras are Bousfield lattices

Given a complete Heyting algebra we construct an algebraic tensor
triangulated category whose Bousfield lattice is the Booleanization of the
given Heyting algebra. As a consequence we deduce that any complete Boolean
algebra is the Bousfield lattice of some tensor triangulated category. Using
the same ideas we then give two further examples illustrating some interesting
behaviour of the Bousfield lattice.Comment: 10 pages, update to clarify the products occurring in the main
constructio

### Mapping spaces in Quasi-categories

We apply the Dwyer-Kan theory of homotopy function complexes in model
categories to the study of mapping spaces in quasi-categories. Using this,
together with our work on rigidification from [DS1], we give a streamlined
proof of the Quillen equivalence between quasi-categories and simplicial
categories. Some useful material about relative mapping spaces in
quasi-categories is developed along the way

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