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Coherence for categorified operadic theories
"... It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) ..."
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Cited by 2 (0 self)
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It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) theory P, and show that every weak Pcategory is equivalent via Pfunctors and Ptransformations to a strict Pcategory. This strictification functor is then shown to have an interesting universal property. 1
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 whose commutativity is ..."
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Cited by 1 (1 self)
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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
"... 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.