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Investigating The Algebraic Structure of Dihomotopy Types
, 2001
"... This presentation is the sequel of a paper published in GETCO'00 proceedings where a research program to construct an appropriate algebraic setting for the study of deformations of higher dimensional automata was sketched. This paper focuses precisely on detailing some of its aspects. The main idea ..."
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Cited by 8 (1 self)
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This presentation is the sequel of a paper published in GETCO'00 proceedings where a research program to construct an appropriate algebraic setting for the study of deformations of higher dimensional automata was sketched. This paper focuses precisely on detailing some of its aspects. The main idea is that the category of homotopy types can be embedded in a new category of dihomotopy types, the embedding being realized by the Globe functor. In this latter category, isomorphism classes of objects are exactly higher dimensional automata up to deformations leaving invariant their computer scientific properties as presence or not of deadlocks (or everything similar or related). Some hints to study the algebraic structure of dihomotopy types are given, in particular a rule to decide whether a statement/notion concerning dihomotopy types is or not the lifting of another statement/notion concerning homotopy types. This rule does not enable to guess what is the lifting of a given notion/statement, it only enables to make the verification, once the lifting has been found.
Weak units and homotopy 3types
, 2006
"... for a weak unit I in an otherwise completely strict monoidal 2category. This implies a version of Simpson’s weakunit conjecture in dimension 3, namely that oneobject 3groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply ..."
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Cited by 6 (2 self)
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for a weak unit I in an otherwise completely strict monoidal 2category. This implies a version of Simpson’s weakunit conjecture in dimension 3, namely that oneobject 3groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3types. The proof has a clear intuitive content and relies on a geometrical argument with string diagrams and configuration spaces. The subtleties and challenges of higher category theory start with the observation (in fact, not a trivial result) that not every weak 3category is equivalent to a strict 3category. The topological counterpart of this is that not every homotopy 3type can be realised by a strict 3groupoid. The discrepancy between the strict and
SEMISTRICT TAMSAMANI NGROUPOIDS AND CONNECTED NTYPES
, 2007
"... Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of co ..."
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Cited by 5 (1 self)
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Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of connected ntypes.
NOTE ON COMMUTATIVITY IN DOUBLE SEMIGROUPS AND TWOFOLD MONOIDAL CATEGORIES
"... A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of EckmannHilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and a ..."
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Cited by 2 (1 self)
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A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of EckmannHilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and all inverse double semigroups are commutative. Stepping up one dimension, the result is used to prove that all strictly associative twofold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative oneobject, onearrow 3groupoids (with weak units) cannot realise all simplyconnected homotopy 3types. 1. Introduction and
unknown title
, 2006
"... Note on commutativity in double semigroups and twofold monoidal categories ..."
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Note on commutativity in double semigroups and twofold monoidal categories
Joachim Kock: Units 20060608 16:46 [1/29] Elementary remarks on units in monoidal categories
, 2006
"... We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1categorical sense). This notion is more economical than the usual notion in terms of leftright constraints, and is motivated by higher category theory ..."
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We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1categorical sense). This notion is more economical than the usual notion in terms of leftright constraints, and is motivated by higher category theory. To start, we describe the semimonoidal category of all possible unit structures on a given semimonoidal category and observe that it is contractible (if nonempty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent nonalgebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is selfcontained. All arguments are elementary, some of them of a certain beauty.
Weak units and homotopy 3types For Ross Street, on his 60th birthday
, 2006
"... We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2category. This implies a version of Simpson’s weakunit conjecture in dimension 3, namely that oneobject 3groupoids that are strict in all respects, except that the object h ..."
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We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2category. This implies a version of Simpson’s weakunit conjecture in dimension 3, namely that oneobject 3groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3types. The proof has a clear intuitive content and relies on a geometrical argument with string diagrams and configuration spaces. 0
Joyal & Kock: Weak units and homotopy 3types [1/21] Weak units and homotopy 3types
, 2006
"... We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2category. This implies a version of Simpson’s weakunit conjecture in dimension 3, namely that oneobject 3groupoids that are strict in all respects, except that the object h ..."
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We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2category. This implies a version of Simpson’s weakunit conjecture in dimension 3, namely that oneobject 3groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3types. The proof has a clear intuitive content and relies on a geometrical argument with string diagrams and configuration spaces. 0
The
, 708
"... periodic table of ncategories for low dimensions I: degenerate categories and degenerate bicategories ..."
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periodic table of ncategories for low dimensions I: degenerate categories and degenerate bicategories