arXiv subjects
Publications and source records attributed to David Blanc.
We describe a comonad on $n$-track categories, for each $n\geq 0$ yielding an explicit cosimplicial abelian group model for the Andr\'{e}-Quillen cohomology of an $(\infty,1)$-category.
We study the homotopy theory of diagrams of chain complexes over a field indexed by a finite poset, and show that it can be completely described in terms of appropriate diagrams of graded vector spaces.
We explain how the simplicial higher-order unstable homotopy operations defined in [BBS2] may be composed and inserted one in another, thus forming a coherent if complicated algebraic structure.
Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the Andr\'{e}-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$ take value. Given a map of simplicial categories $\phi:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of $\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to study the obstruction to lifting $\phi$ to $P^{(n)}\mathcal{X}$. In particular, an explicit description of this obstruction for the boundary of a cube can be used to recover various higher homotopy invariants of $\mathcal{X}$.
A $configuration$ of a linkage $\Gamma$ is a possible positioning of $\Gamma$ in $\mathbb{R}^d$ and the collection of all such forms the configuration space $\mathcal{C}(\Gamma)$ of $\Gamma$. We here introduce the notion of the $symmetric configuration space$ of a linkage, in which we identify configurations which are geometrically indistinguishable. We show that the symmetric configuration space of a planar polygon has a regular cell structure, provide some principles for calculating this structure, and give a complete description of the symmetric configuration space of all quadrilaterals and of the equilateral pentagon.
We describe two types of localization for $(\infty, 1)$-categories which determine the successive terms in the homotopy spectral sequence of a (co)simplicial object.
We explain how to set up the homotopy spectral sequence of a (co)simplicial object in an $\infty$-category, with an emphasis on how to construct the differentials in a model-invariant manner.
The $E_2$-term of the Adams spectral sequence for $\mathbf{Y}$ may be described in terms of its cohomology $E^\ast \mathbf{Y}$, together with the action of the primary operations $E^\ast \mathbf{E}$ on it, for ring spectra such as $\mathbf{E} = \mathbf{H}\mathbb{F}_p$. We show how the higher terms of the spectral sequence can be similarly described in terms of the higher order truncated $\mathbf{E}$-mapping algebra for $\mathbf{Y}$ $\; - \;$ that is truncations of the function spectra $\operatorname{Fun}(\mathbf{Y}, \mathbf{M})$ for various $\mathbf{E}$-modules $\mathbf{M}$, equipped with the action of $\operatorname{Fun}(\mathbf{M}, \mathbf{M}')$ on them.
We provide a general definition of Toda brackets in a pointed model categories, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations.
We construct certain unstable higher-order homotopy operations indexed by the simplex categories of $\Delta^{n}$ for ${n\geq 2}$ and prove that all elements in the homotopy groups of a wedge of spheres are generated under such operations by Whitehead products and the group structure. This provides a stronger unstable analogue of Cohen's theorem on the decomposition of stable homotopy.
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $X$ up to weak equivalence. It can also be used to distinguish between different maps $f$ from $X$ to $Y$ which induce the same morphism on homotopy groups $f_*$ from $\pi_* X$ to $\pi_* Y$.
We prove two representability theorems, up to homotopy, for presheaves taking values in a closed symmetric combinatorial model category \cat V. The first theorem resembles the Freyd representability theorem, the second theorem is closer to the Brown representability theorem. As an application we discuss a recognition principle for mapping spaces.
We define a comonad cohomology of track categories and we show it is linked by a long exact sequence to its Dwyer-Kan-Smith cohomology . Under mild hypothesis on the track category, we show that its comonad cohomology coincides, up to dimension shift, with its Dwyer-Kan-Smith cohomology, therefore obtaining an algebraic formulation of the latter. We also specialize our results to the case where the track category is a $2$-groupoid.
In this paper we provide an explicit general construction of higher homotopy operations in model categories, which include classical examples such as (long) Toda brackets and (iterated) Massey products, but also cover unpointed operations not usually considered in this context. We show how such operations, thought of as obstructions to rectifying a homotopy-commutative diagram, can be defined in terms of a double induction, yielding intermediate obstructions as well.
Let $R=\mathbb{F}_p$ or a field of characteristic $0$. For each $R$-good topological space $Y$, we define a collection of higher cohomology operations which, together with the cohomology algebra $H^*(Y;R)$ suffice to determine $Y$ up to $R$-completion. We also provide a similar collection of higher cohomology operations which determine when two maps $f_0,f_1: Z\to Y$ between $R$-good spaces(inducing the same algebraic homomorphism $H^*(Y;R)\to H^*(Z;R)$) are $R$-equivalent.
We describe a variant construction of the unstable Adams spectral the sequence for a space $Y$, associated to any free simplicial resolution of $H^*(Y;R)$ for $R=\mathbb{F}_p$ or $\mathbb{Q}$. We use this construction to describe the differentials and filtration in the spectral sequence in terms of appropriate systems of higher cohomology operations.
We construct a completed version C(Gamma) of the configuration space of a linkage Gamma in R^3 which takes into account the ways one link can touch another. We also describe a simplified version of C(Gamma) which is a blow-up of the space of immersions of Gamma in R^3 A number of simple detailed examples are given.
We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and chain complexes and Massey products for differential graded algebras, among others.