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Double categories, 2-categories, thin structures and connections. (English) Zbl 0918.18005

This paper is concerned with structures on double categories in which the horizontal and vertical edge categories coincide (edge symmetric double categories). A commutative square of edge morphisms in such a double category is a quadruple \((a,b,c,d)\) of edge morphisms such that \(ab=cd\), and a filler for a commutative square \((a,b,c,d)\) is a \(2\)-morphism with \(a\), \(b\), \(c\) and \(d\) as its sources and targets. A thin structure is a functor assigning a filler to each commutative square. A connection is a function, satisfying certain natural conditions, which assigns fillers to commutative squares of the forms \((a,1,a,1)\) and \((1,b,1,b)\). The main result is that thin structures and connections are equivalent. It is also shown that edge symmetric double categories with either structure are equivalent to \(2\)-categories.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
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