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A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
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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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.