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30
Combinatorics Of Branchings In Higher Dimensional Automata
 Theory Appl. Categ
, 2001
"... We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory ca ..."
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Cited by 35 (9 self)
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We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory called the reduced branching homology. The latter is the homology of the quotient of the branching complex by the subcomplex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is #categories freely generated by precubical sets. As application, we calculate the branching homology of some #categories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side.
Diagonals on the Permutahedra, Multiplihedra and associahedra
 J. HOMOLOGY, HOMOTOPY AND APPL
, 2004
"... We construct an explicit diagonal ∆P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks ’ projection P → K [19] and its factorization through J. We introduce the notion of a permutahedral set Z and lift ∆P to a diagonal on Z. We show that the ..."
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Cited by 31 (7 self)
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We construct an explicit diagonal ∆P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks ’ projection P → K [19] and its factorization through J. We introduce the notion of a permutahedral set Z and lift ∆P to a diagonal on Z. We show that the double cobar construction Ω²C∗(X) is a permutahedral set; consequently ∆P lifts to a diagonal on Ω²C∗(X). Finally, we apply the diagonal on K to define the tensor product of A∞(co)algebras in maximal generality.
The compression theorem
 I., Geom. Topol
"... This the first of a set of three papers about the Compression Theorem: if M m is embedded in Q q × R with a normal vector field and if q − m ≥ 1, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q × R. The theorem can ..."
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Cited by 20 (9 self)
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This the first of a set of three papers about the Compression Theorem: if M m is embedded in Q q × R with a normal vector field and if q − m ≥ 1, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q × R. The theorem can be deduced from Gromov’s theorem on directed embeddings [5; 2.4.5 (C ′)] and is implicit in the preceeding discussion. Here we give a direct proof that leads to an explicit description of the finishing embedding. In the second paper in the series we give a proof in the spirit of Gromov’s proof and in the third part we give applications.
The generalized Baues problem
, 1998
"... Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivat ..."
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Cited by 17 (0 self)
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Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to triangulations, zonotopal tilings, monotone paths in linear programming, oriented matroid Grassmannians, singularities, and homotopy theory. Included are several open questions and problems. 1.
A canonical enriched AdamsHilton model for simplicial sets, preprint arXiv:math.AT/0408216
"... Abstract. For any 1reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that Szczarba’s canonical quasiisomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the lo ..."
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Cited by 11 (11 self)
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Abstract. For any 1reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that Szczarba’s canonical quasiisomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the loop group of K, is a strongly homotopy coalgebra map.
From Operads to `Physically' Inspired Theories
"... Introduction As evidenced by these conferences (Hartford and Luminy), operads have had a renaissance in recent years for a variety of reasons. Originally studied entirely as a tool in homotopy theory, operads have recently received new inspirations from homological algebra, category theory, algebra ..."
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Cited by 11 (1 self)
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Introduction As evidenced by these conferences (Hartford and Luminy), operads have had a renaissance in recent years for a variety of reasons. Originally studied entirely as a tool in homotopy theory, operads have recently received new inspirations from homological algebra, category theory, algebraic geometry and mathematical physics. I'll try to provide a transition from the foundations to the frontier with mathematical physics. For me, the transition occurred in two stages. First, there is the generalization of Lie algebra cohomology known as BRST (BecchiRouetStora Tyutin) cohomology, which turned out to be very closely related to strong homotopy Lie (L1 ) algebras, which I will describe later in homological algebraic terms  along the lines of Balavoine's talk at this conference [5]. That description makes no use of operads, but the relevance of operads appeared later in the work of Hinich and Schechtman [25]. Operads rev
A cubical model of a fibration
 J. Pure Appl. Algebra
"... Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products f ..."
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Cited by 8 (7 self)
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Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products for homotopy Galgebras allows to obtain multiplicative models for fibrations. 1.
Corings over operads characterize morphisms, math.AT/0505559
"... objects in M, with its composition monoidal structure. Let R be a Pcoring, ..."
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Cited by 7 (1 self)
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objects in M, with its composition monoidal structure. Let R be a Pcoring,
Categories of spectra and infinite loop spaces
 Lecture Notes in Mathematics Vol. 99, SpringerVerlag
, 1969
"... I presented a calculation of H,(F;Zp) as an algebra, for odd primes p, where F = lim F(n) and F(n) is the topological monoid of homotopy equivalences of an nsphere. This computation was meant as a preliminary step towards the computation of H*(BF;Zp). Since then, I have calculated H*(BF;Zp), for a ..."
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Cited by 6 (4 self)
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I presented a calculation of H,(F;Zp) as an algebra, for odd primes p, where F = lim F(n) and F(n) is the topological monoid of homotopy equivalences of an nsphere. This computation was meant as a preliminary step towards the computation of H*(BF;Zp). Since then, I have calculated H*(BF;Zp), for all primes p, as a Hopf algebra over the Steenrod and DyerLashof algebras. The calculation, while not difficult, is somewhat lengthy, and I was not able to write up a coherent presentation in time for inclusion in these proceedings. The computation required a systematic study of homology operations on nfold and infinite loop spaces. As a result of this study, I have also been able to compute H,(2nsnx;Zp), as a Hopf algebra over the Steenrod algebra, for all connected spaces X and prime numbers p. This result, which generalizes those of Dyer and Lashof [3] and Milgram [8], yields explicit descriptions of both H,(~nsnx;Zp) and H,(QX;Zp), QX = li~> 2nsnx, as functors of H,(X;Zp). An essential first step towards these results was a systematic categorical analysis of the notions of nfold and infinite loop spaces. The results of this analysis will 449 be presented here. These include certain adjoint functor relationships that provide the conceptual reason that H.(~nsnx;zp) and H,(QX;Zp) are functors of H,(X;Zp) and that precisely relate maps between spaces to maps between spectra. These categorical considerations motivate the introduction of certain nonstandard categories, I and i, of (bounded) spectra and ~spectra, and the main purpose of this paper is to propagandize these categories. It is clear from their definitions that these categories are considerably easier to work with topologically than are the usual ones, but it is not clear that they are sufficiently large to be of interest. We shall remedy this by showing that, in a sense to be made precise, these categories are equivalent for the purposes of homotopy theory to the standard categories of (bounded) spectra and ~spectra. We extend the theory to unbounded spectra in the last section.