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The EckmanHilton argument and higher operads, preprint
, 2006
"... To the memory of my father. 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 ..."
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Cited by 3 (2 self)
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To the memory of my father. 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 begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the EckmanHilton type argument in terms of the calculation of left Kan extensions in an appropriate 2category. Then we apply this scheme to the case of noperads in the author’s sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of noperads to the
Higherdimensional Mac Lane's pentagon and Zamolodchikov equations
, 1999
"... An important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that \all diagrams commute". This paper gives a higherdimensional generalization of Mac Lane's pentagon: a 6dimensional diagram who ..."
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An important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that \all diagrams commute". This paper gives a higherdimensional generalization of Mac Lane's pentagon: a 6dimensional diagram whose commutativity is needed in order for all diagrams in somewhat weak teisi to commute. Looping twice gives a 4dimensional diagram in somewhat weak braided teisi, of which ve 3dimensional edges can be interpreted as proofs of ve dierent Zamolodchikov equations in braided monoidal 2categories. Hence higherdimensional Mac Lane's pentagon expresses the relations between these proofs concisely. 1 Introduction The coherence theorem for tricategories states that every tricategory is triequivalent to a Graycategory [6]. But there is also another coherence theorem for tricategories, stating that tricategories are (algebras for a) contractible (operad) [1], which roughly says that \all diagrams in a tricategory...
unknown title
, 2006
"... Let R be a ring and M a bimodule over R. Then there are three essential cohomology theories associated to the pair (R, M) due to Hochschild, Shukla, and ..."
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Let R be a ring and M a bimodule over R. Then there are three essential cohomology theories associated to the pair (R, M) due to Hochschild, Shukla, and
unknown title
, 2006
"... Abstract. MacLane cohomology is an algebraic version of the topological Hochschild cohomology. Based on the computation of the third author (see Appendix below) we obtain an interpretation of the third Mac Lane cohomology of rings using certain kind of crossed extensions of rings in the quadratic wo ..."
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Abstract. MacLane cohomology is an algebraic version of the topological Hochschild cohomology. Based on the computation of the third author (see Appendix below) we obtain an interpretation of the third Mac Lane cohomology of rings using certain kind of crossed extensions of rings in the quadratic world. Actually we obtain two such interpretations corresponding to the two monoidal structures on the category of square groups.
unknown title
, 2006
"... Let R be a ring and M a bimodule over R. Then there are three essential cohomology theories associated to the pair (R, M) due to Hochschild, Shukla, and ..."
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Let R be a ring and M a bimodule over R. Then there are three essential cohomology theories associated to the pair (R, M) due to Hochschild, Shukla, and
This document in subdirectoryNS/98/7/ Contents 2Categories
, 909
"... See back inner page for a list of recent BRICS Notes Series publications. Copies may be obtained by contacting: BRICS ..."
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See back inner page for a list of recent BRICS Notes Series publications. Copies may be obtained by contacting: BRICS