The formula is $\partial e=\left(a{d}_e\right)b+{\sum }_{i=0}^{\infty }\left(B_i\right)/i!{\left(a{d}_e\right)}^i(b-a)$, with ∂a + 1/2 [a,a] = 0 and ∂b + 1/2 [b,b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a,b,e]. The coefficients are defined by $x/(e^x-1)={\sum }_{n=0}^{\infty }Bₙ/n!xⁿ$. The theorem is that ∙ this formula for ∂ on generators extends to a derivation of square zero on L[a,b,e]; ∙ the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the “flow” generated by e moves a to b in unit...

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of $tow\left(H_0\right)$ is an isomorphism if Y is movable. Recall that $\left(H_0\right)$ is the full subcategory of $pro-H_0$ consisting of...

We study matrix factorizations of a potential W which is a section of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.

We prove that the space $ex{p}_k\bigvee S^{m+1}$ of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space $ex{p}_{k}X$ is (m+k-2)-connected for any m-connected cell complex X.

In questo articolo studiamo i gruppi ${\mathrm{\Theta }}_h^{\mathcal{F}}$ di una sfera $S^n$ e proviamo che il gruppo ${\mathrm{\Theta }}_n^{\mathcal{F}}\left(S^n,x_0\right)$ è isomorfo all'ennesimo gruppo di omotopia di $\left(S^n,x_0\right)$, nell'ipotesi che $\mathcal{F}$ sia una classe coconnessa di links ammissibili.

We consider Taylor approximation for functors from the small category of finite pointed sets $\Gamma$ to modules and give an explicit description for the homology of the layers of the Taylor tower. These layers are shown to be fibrant objects in a suitable closed model category structure. Explicit calculations are presented in characteristic zero including an application to higher order Hochschild homology. A spectral sequence for the homology of the homotopy fibres of this approximation is provided.

We study the Taylor towers of the nth symmetric and exterior power functors, Spⁿ and Λⁿ. We obtain a description of the layers of the Taylor towers, $D_{k}Spⁿ$ and $D_k\Lambda ⁿ$, in terms of the first terms in the Taylor towers of $S{p}^t$ and ${\Lambda }^t$ for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of $S{p}^t$ and ${\Lambda }^t$. We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for $D_{k}Spⁿ$ and $D_k\Lambda ⁿ$.

The suspension and loop space functors, Σ and Ω, operate on the lattice of Bousfield classes of (sufficiently highly connected) topological spaces, and therefore generate a submonoid ℒ of the complete set of operations on the Bousfield lattice. We determine the structure of ℒ in terms of a single parameter of homotopy theory which is closely tied to the problem of desuspending weak cellular inequalities.