Results 1  10
of
23
Homotopy Invariants of Higher Dimensional Categories and Concurrency in Computer Science
, 1999
"... The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other one ..."
Abstract

Cited by 49 (10 self)
 Add to MetaCart
The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of C. Two natural linear maps called the negative (resp. the positive) Hurewicz morphism from the globular homology to the negative (resp. positive) corner homology are constructed. We explain the reason why these constructions allow the reinterpretation of some geometric problems coming from computer science.
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 ..."
Abstract

Cited by 35 (9 self)
 Add to MetaCart
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.
The EckmannHilton argument, higher operads and Enspaces, available at http://www.ics.mq.edu.au
 mbatanin/papers.html of Homotopy and Related Structures
"... The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2category, then its Homset is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an noperad A in the author’s sense there exists a symmetric operad S n (A) called the nfold suspension of A such that the
A Cellular Nerve for Higher Categories
, 2002
"... ... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of Aalgebras for each ooperad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of Aalgebras (i.e. weak ocategories) is
Cubical Sets And Their Site
 Theory Appl. Categ
, 2003
"... Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicia ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid ; in fact, K is the classifying category of a monoidal algebraic theory of such monoids. Analogous results are given for the restricted cubical site I of ordinary cubical sets (just faces and degeneracies) and for the intermediate site J (including connections). We also consider briefly the reversible analogue, !K.
Iterated wreath product of the simplex category and iterated loop spaces
 Adv. Math
"... Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternat ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γspace. In particular, each EilenbergMacLane space has a canonical reduced Θnset model. The number of (n + d)dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.
Computads for Finitary Monads on Globular Sets
, 1998
"... . A finitary monad A on the category of globular sets provides basic algebraic operations from which more involved `pasting' operations can be derived. To makes this rigorous, we define Acomputads and construct a monad on the category of Acomputads whose algebras are Aalgebras; an action of the n ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
. A finitary monad A on the category of globular sets provides basic algebraic operations from which more involved `pasting' operations can be derived. To makes this rigorous, we define Acomputads and construct a monad on the category of Acomputads whose algebras are Aalgebras; an action of the new monad encapsulates the pasting operations. When A is the monad whose algebras are ncategories, an Acomputad is an ncomputad in the sense of R.Street. When A is associated to a higher operad (in the sense of the author) , we obtain pasting in weak ncategories. This is intended as a first step towards proving the equivalence of the various definitions of weak ncategory now in the literature. Introduction This work arose as a reflection on the foundation of higher dimensional category theory. One of the main ingredients of any proposed definition of weak ncategory is the shape of diagrams (pasting scheme) we accept to be composable. In a globular approach [3] each kcell has a source ...
The branching nerve of HDA and the Kan condition
 Theory and Applications of Categories
, 2003
"... One can associate to any strict globular omegacategory three augmented simplicial nerves called the globular nerve, the branching and the merging semicubical nerves. If this strict globular omegacategory is freely generated by a precubical set, then the corresponding homology theories contain dif ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
One can associate to any strict globular omegacategory three augmented simplicial nerves called the globular nerve, the branching and the merging semicubical nerves. If this strict globular omegacategory is freely generated by a precubical set, then the corresponding homology theories contain different informations about the geometry of the higher dimensional automaton modeled by the precubical set. Adding inverses in this omegacategory to any morphism of dimension greater than 2 and with respect to any composition laws of dimension greater than 1 does not change these homology theories. In such a framework, the globular nerve always satisfies the Kan condition. On the other hand, both branching and merging nerves never satisfy it, except in some very particular and uninteresting situations. In this paper, we introduce two new nerves (the branching and merging semiglobular nerves) satisfying the Kan condition and having conjecturally the same simplicial homology as the branching and merging semicubical nerves respectively in such framework. The latter conjecture is related to the thin elements conjecture already introduced in our previous papers.
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences