It is shown that a reduced homology theory on the category of pointed compact metric spaces is strong shape invariant if and only if its homology functors hn satisfy the quotient exactness axiom, which means that for each pointed compact metric pair (X, A, a0) the natural sequence hn(A, a0) → hn(X, a0) → hn(X/A, *) is exact. As a consequence, all generalized Steenrod homology theories are strong shape invariant.

Let $X$ be a 2-connected $p$-local finite $H$-space with a single cell in dimension three. We give a simple cohomological criterion which distinguishes when the inclusion i: $S^3 \underset {\longrightarrow}{i} X$ has the property that the loop of its three-connected cover is null homotopic. In particular, such a null homotopy implies that $\pi_m(i )= 0$ for $m \geq 4$. Applications are made to Harper's rank 2 finite $H$-space and simple, simply-connected, compact Lie groups.

Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a monad $\top$ on a simplicial category $\mathcal{C}$, we instead show how s.h. $\top$-algebras over $\mathcal{C}$ naturally form a Segal space. Given a distributive monad-comonad pair ($\top, \bot$), the same is true for s.h. ($\top, \bot$)-bialgebras over $\mathcal{C}$; in particular this yields the homotopy theory of s.h. sheaves of s.h. rings. There are similar statements for quasi-monads and quasi-comonads. We also show how the structures arising are related to derived connections on bundles.

Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.

In this article, we associate new invariants called pseudo-homology and pseudo-cohomology groups to right quasigroups and in turn to geometric spaces with natural quasigroup structures and to general extensions. It initiates a program to differentiate different types of general extensions.

We give an algebraic proof for the result of Eilenberg and Mac Lane that the second cohomology group of a simplicial group G can be computed as a quotient of a fibre product involving the first two homotopy groups and the first Postnikov invariant of G. Our main tool is the theory of crossed module extensions of groups.

This paper studies the category of posets Pos as a model for the homotopy theory of spaces. We prove that: (i) Pos admits a (cofibrantly generated and proper) model structure and the inclusion functor Pos → Cat into Thomason's model category is a right Quillen equivalence, and (ii) there is a proper class of different choices of cofibrations for a model structure on Pos or Cat where the weak equivalences are defined by the nerve functor. We also discuss the homotopy theory of posets from the viewpoint of Alexandroff T0-spaces, and we apply a result of McCord to give a new proof of the classification theorems of Moerdijk and Weiss in the case of posets.

Path categories are defined, and their basic properties are described, for simplicial and cubical sets. A calculational method for describing the path category P(K) of a finite oriented simplicial complex K is introduced, which involves a finite 2-category which can be specified by generators and relations. This method specializes to higher dimensional automata via the triangulation functor from cubical to simplicial sets, and leads to calculations of their associated execution paths.

We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.

The cyclic, periodic cyclic and negative cyclic homologies of associative algebras are fitted into the context of cotriple homology of Barr and Beck. As applications of these results, an axiomatic description of the cyclic homology theory and the Hopf type formulas in the sense of Brown-Ellis are given.

We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.

We compute the cohomology $H^*(\mathcal{H},k)=\rm{Ext}^*_\mathcal{h}(k,k) $where $\mathcal{H} = \mathcal{H} (n,q)$ is the Hecke algebra of the symmetric group $\mathfrak{S}_n$ at a primitive $\ell$th root of unity $q$, and $k$ is a field of characteristic zero. The answer is particularly interesting when $\ell= 2$, which is the only case where it is not graded commutative. We also carry out the corresponding computation for Hecke algebras of type $B_n$ and $D_n$ when $\ell$ is odd.

Let Aut($p$) denote the space of all self-fibre-homotopy equivalences of a fibration $p : E \to B$. When $E$ and $B$ are simply connected CW complexes with $E$ finite, we identify the rational Samelson Lie algebra of this monoid by means of an isomorphism: $$\pi_*(\rm{Aut} (p)) \otimes \mathbb{Q} \cong H_*(\rm{Der}_{\wedge V}(\wedge V \otimes \wedge W)).$$

Here $\wedge V \to \wedge V \otimes \wedge W$ is the Koszul-Sullivan model of the fibration and $\rm{Der}_{\wedge V} (\wedge{V} \otimes \wedge W)$ is the DG Lie algebra of derivations vanishing on $\wedge V$. We obtain related identifications of the rationalized homotopy groups of fibrewise mapping spaces and of the rationalization of the nilpotent group $\pi_0 (\rm{Aut}_\sharp(p)$ is a fibrewise adaptation of the submonoid of maps inducing the identity on homotopy groups.