Ivancevic_Applied-Diff-Geom

134 Applied Differential Geometry: A Modern Introductionwhere all the arrows are 2−morphisms built using the associator. Weak3−categories or tricategories were defined by [Gordon et. al. (1995)].In a tricategory, the pentagon identity holds only up to an invertible3−morphism, which satisfies a further coherence law of its own.When one explicitly lists the coherence laws this way, the definitionof weak n−category tends to grow ever more complicated with increasingn. To get around this, one must carefully study the origin of these coherencelaws. So far, most of our insight into coherence laws has been wonthrough homotopy theory, where it is common to impose equations onlyup to homotopy, with these homotopies satisfying coherence laws, again upto homotopy, and so on. For example, the pentagon identity and highercoherence laws for associativity first appeared in Stasheff’s work on thestructure inherited by a space equipped with a homotopy equivalence to aspace with an associative product [Stasheff (1963)]. Subsequent work ledto a systematic treatment of coherence laws in homotopy theory throughthe formalism of topological operads [Adams (1978)].Underlying the connection between homotopy theory and n−categorytheory is a hypothesis made quite explicit by Grothendieck [Grothendieck(1983)]: to any topological space one should be able to associatean n−category having points as objects, paths between points as1−morphisms, certain paths of paths as 2−morphisms, and so on, withcertain homotopy classes of n−fold paths as n−morphisms. This shouldbe a special sort of weak n−category called a weak n−groupoid, in whichall j−morphisms (0 < j ≤ n) are equivalences. Moreover, the processof assigning to each space its fundamental n−groupoid, as Grothendieckcalled it, should set up a complete correspondence between the theory ofhomotopy n−types (spaces whose homotopy groups vanish above the nth)and the theory of weak n−groupoids. This hypothesis explains why all thecoherence laws for weak n−groupoids should be deducible from homotopytheory. It also suggests that weak n−categories will have features not foundin homotopy theory, owing to the presence of j−morphisms that are notequivalences [Baez and Dolan (1998)].Homotopy theory also makes it clear that when setting up a theoryof n−categories, there is some choice involved in the shapes of onesj−morphisms – or in the language of topology, j−cells. The traditionalapproach to n−cate-gories is globular. This means that for j > 0, eachj−cell f : x → y has two (j − 1)−cells called its source, sf = x, and target,