15,658 research outputs found
A Linear Category of Polynomial Diagrams
We present a categorical model for intuitionistic linear logic where objects
are polynomial diagrams and morphisms are simulation diagrams. The
multiplicative structure (tensor product and its adjoint) can be defined in any
locally cartesian closed category, whereas the additive (product and coproduct)
and exponential Tensor-comonoid comonad) structures require additional
properties and are only developed in the category Set, where the objects and
morphisms have natural interpretations in terms of games, simulation and
strategies.Comment: 20 page
Calculus of functors, operad formality, and rational homology of embedding spaces
Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the
homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the
rational homology of Ebar(M,V). We study it by applying embedding calculus and
orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main
theorem states that if the dimension of V is more than twice the embedding
dimension of M, the Taylor tower in the sense of orthogonal calculus
(henceforward called ``the orthogonal tower'') of this functor splits as a
product of its layers. Equivalently, the rational homology spectral sequence
associated with the tower collapses at E^1. In the case of knot embeddings,
this spectral sequence coincides with the Vassiliev spectral sequence. The main
ingredients in the proof are embedding calculus and Kontsevich's theorem on the
formality of the little balls operad.
We write explicit formulas for the layers in the orthogonal tower of the
functor HQ /\Ebar(M,V)_+. The formulas show, in particular, that the (rational)
homotopy type of the layers of the orthogonal tower is determined by the
(rational) homology type of M. This, together with our rational splitting
theorem, implies that under the above assumption on codimension, the rational
homology groups of Ebar(M,V) are determined by the rational homology type of M.Comment: 35 pages. An erroneous definition in the last section was corrected,
as well as several misprints. The introduction was somewhat reworked. The
paper was accepted for publication in Acta Mathematic
The universal Vassiliev invariant for the Lie superalgebra gl(1|1)
We compute the universal weight system for Vassiliev invariants coming from
the Lie superalgebra gl(1|1) applying the construction of \cite{YB}. This
weight system is a function from the space of chord diagrams to the center
of the universal enveloping algebra of gl(1|1), and we find a combinatorial
expression for it in terms of the standard generators of . The resulting
knot invariants generalize the Alexander-Conway polynomial.Comment: 44 pages with figures, wrapped with uufiles, requires epsf.sty --
Added a short section about deframin
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