This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for...

The Grothendieck construction is a process to form a single category from a diagram of small categories. In this paper, we extend the definition of the Grothendieck construction to diagrams of small categories enriched over a symmetric monoidal category satisfying certain conditions. Symmetric monoidal categories satisfying the conditions in this paper include the category of k-modules over a c...

We show that the topological Hochschild homology THH(R) of an En-ring spectrum R is an En−1-ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the the little n-cubes operad Cn is an En+1-operad, a result which is of independent interest. In 1993 Deligne asked whether the Hochschild cochain complex of an associative ring has a canoni...

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces based topological spaces. We construct between complexification-realification adjunction These allow for movement versions calculi, comparisons Taylor towers produced by both calculi. show that when inputted functor is weakly polynomial, tower restricted through realifica...

One of the important tools for working with the classifying toposes is Diaconescu's theorem, describing geometric morphisms to the topos of presheaves on a small category C in terms of at functors on C. In 4], this theorem is generalized from discrete categories to topological categories, with the additional requirement that the map assigning to an arrow its source is etale (a local homeomorphi...

Elmendorf’s Theorem in equivariant homotopy theory states that for any topological group G, the model category of G-spaces is Quillen equivalent to the category of continuous diagrams of spaces indexed by the opposite of the orbit category of G with the projective model structure. For discrete G, Bert Guillou explored equivariant homotopy theory for any cofibrantly generated model category C an...

Contour, split and join trees can be defined as functors acting on the category of scalar fields, whose morphisms are value-preserving functions. The categorical definition provides a natural way to efficiently compute a variety of topological properties of all contours, sublevel or superlevel components in a scalar field. The result is a labeling of the contour, split or join tree and can be u...

Abstract We construct natural operators connecting the cohomology of moduli spaces stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover topological mirror symmetry conjecture Hausel Thaddeus concerning $\mathrm {SL}_n$ - {PGL}_n$ -Higgs bundles. This provides a complete description space in terms tautological classes, gives new proof Hausel–Thaddeu...

We define natural A∞-transformations and construct A∞-category of A∞-functors. The notion of non-strict units in an A∞-category is introduced. The 2-category of (unital) A∞-categories, functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta63, II] Stasheff def...