Using simplicial machinery, which is analogous to that developed by Bousfield-Kan, we describe a first quadrant homotopy spectral sequence for a termwise connected pointed simplicial space X in terms of differential relations on the normalized homotopy of X and establish the existence of a smash product pairing.

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

Let $G$ be a finite group and $mathbb{Z}[G]$ its integral group ring. We prove that the nil groups $N^j K_2 \mathbb{Z}[G])$ do not vanish for all $j \geq 1$ and for a large class of finite groups. We obtain from this that the iterated nil groups $N^j K_i (\mathbb{Z}[G])$ are also nonzero for all $i \geq 2, j \geq i - 1$

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String($n$). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the 'Jacobiator.' Similarly, a Lie 2-group is a categorified version of a Lie group. If $G$ is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras $\mathfrak{g}_k$ each having $\mathfrak{g}$ as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on $G$. There appears to be no Lie 2-group having $\mathfrak{g}_\mathcal{k}$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral $k$ an infinite-dimensional Lie 2-group $\mathcal{P}_k G$ whose Lie 2-algebra is equivalent to $\mathfrak{g}_\mathcal{k}$. The objects of $\mathfrak{g}_\mathcal{k}$ are based paths in $G$-, while the automorphisms of any object form the level-$k$ Kac-Moody central extension of the loop group $\Omega{G}$. This 2- group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group $|\mathcal{P}_k G|$ that is an extension of $G$ by $K(\mathbb{Z}, 2)$. When $k = \pm 1, |\mathcal{P}_k G|$ can also be obtained by killing the third homotopy group of $G$. Thus, when $G = \rm{Spin}(n), |PkG|$ is none other than String($n$).

Given a functor $T : \mathcal{C} \to \mathcal{D}$ carrying a class of morphisms $S \subset \mathcal C$ into a class $S^\prime \subset \mathcal{D}$, we give sufficient conditions for $T$ to induce an equivalence on the localised categories. These conditions are in the spirit of Quillen’s Theorem A. We give some applications in algebraic and birational geometry.

The erroneous statement (HHA 5 (2003), no. 1, 1–48) that the collection of unital $A_\infty$-categories, all $A_\infty$-functors, and all $A_\infty$-transformations (resp. equivalence classes of natural $A_\infty$- transformations) form a $\mathcal{K}-2$-category $\mathcal{K}^u A_\infty$(resp. ordinary 2-category $^u A_\infty$) is corrected as follows. All 2-category axioms are satisfied, except that $1_e \cdot f$ does not necessarily equal $1_{ef}$ for all composable 1-morphisms $e, f$. The axiom $e \cdot 1_f = 1_{ef}$ does hold. The mistake does not affect results on invertible 2-morphisms and quasi-invertible 1-morphisms in $^u A_\infty$.

We classify thick subcategories of the bounded derived category of a hereditary abelian category A in terms of subcategories of A. The proof can be applied to characterize the localizing subcategories of the full derived category of A. As an application we prove an algebraic analog of the telescope conjecture for the derived category of a representation finite hereditary artin algebra.

Associated to each finite subgroup $\Gamma$ of ${\tt SL}_2(\mathbb{C})$ there is a family of noncommutative algebras $O^\tau(\Gamma)$, which is a deformation of the coordinate ring of the Kleinian singularity $\mathbb{C}^2/\Gamma$. We study finitely generated projective modules over these algebras. Our main result is a bijective correspondence between the set of isomorphism classes of rank one projective modules over $O^\tau$ and a certain class of quiver varieties associated to $\Gamma$s. We show that this bijection is naturally equivariant under the action of a “large” Dixmier-type automorphism group $G$. Our construction leads to a completely explicit description of ideals of the algebras $O^\tau$ .

Let $EK$ be the simplicial suspension of a pointed simplicial set $K$. We construct a chain model of the James map, $\alpha_K : CK \to \Omega CEK$. We compute the cobar diagonal on $\Omega CEK$, not assuming that $EK $is 1-reduced, and show that $\alpha_K$ is comultiplicative. As a result, the natural isomorphism of chain algebras $TCK \cong \Omega CK$ preserves diagonals. In an appendix, we show that the Milgram map, $\Omega (A \otimes B) \to \Omega A \otimes \Omega B$, where $A$ and $B$ are coaugmented coalgebras, forms part of a strong deformation retract of chain complexes. Therefore, it is a chain equivalence even when $A$ and $B$ are not 1-connected.

Let $\mathcal{E}$ be a Frobenius category. Let $\underset {=} {\mathcal{E}}$ denote its stable category. The shift functor on $\underline {E}$ induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in $\underset {=} {\mathcal{E}}$. Shifting a complex by 3 positions yields an outer shift functor on this category. Passing to quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomorphic, via an isomorphism satisfying yet a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a Verdier triangulation on $\underset {=} {\mathcal{E}}$, except for the octahedral axiom. We generalise the notion of acyclic complexes such that the accordingly enlarged version of Heller’s construction includes octahedra.

A subset $P$ of $mathbb{R}^n$ gives rise to a formal Laurent series with monomials corresponding to lattice points in $P$ . Under suitable hypotheses, this series represents a rational function $R(P)$; this happens, for example, when $P$ is bounded in which case $R(P)$ is a Laurent polynomial. Michel Brion [2] has discovered a surprising formula relating the Laurent polynomial $R(P)$ of a lattice polytope $P$ to the sum of rational functions corresponding to the supporting cones subtended at the vertices of $P$ . The result is re-phrased and generalised in the language of cohomology of line bundles on complete toric varieties. Brion's formula is the special case of an ample line bundle on a projective toric variety. The paper also contains some general remarks on the cohomology of torus-equivariant line bundles on complete toric varieties, valid over arbitrary commutative ground rings.

Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach to this problem. We assume that the data is randomly sampled from an unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using statistical estimators for samples from certain families of distributions, we show that we can recover the persistent homology of the underlying distribution.

We describe a minimal unstable module presentation over the Steenrod algebra for the odd-primary cohomology of infinite-dimensional complex projective space and apply it to obtain a minimal algebra presentation for the cohomology of the classifying space of the infinite unitary group. We also show that there is a unique Steenrod module structure on any unstable cyclic module that has dimension one in each complex degree (half the topological degree) with a fixed alpha-number (sum of 'digits') and is zero in other degrees.

We generalize Berger and Moerdijk's results on axiomatic homotopy theory for operads to the setting of enriched symmetric monoidal model categories, and show how this theory applies to orthogonal spectra. In particular, we provide a symmetric fibrant replacement functor for the positive stable model structure.

Cofibrations are defined in the category of Frölicher spaces by weakening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deformation retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Frölicher subspace A of the Frölicher space $X$ is a smooth neighborhood deformation retract of $X$ if and only if the inclusion $i : A \hookrightarrow X$ comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence.

We work over an arbitrary ring $R$. Given two truncated projective resolutions of equal length for the same module, we consider their underlying chain complexes. We show they may be stabilized by projective modules to obtain a pair of complexes of the same homotopy type.

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show that this (co)homology, called Hopf-Hochschild (co)homology, can also be defined as a derived functor on the category of representations of an equivariant analogue of the enveloping algebra of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf-Hochschild (co)homology.