The unicity of the homotopy theory of higher categories.

Higher categories are playing an increasingly important role in algebraic topology and mathematics more generally. Due to their diverse origins there are many competing approaches to the theory. In this talk I will describe joint work with Clark Barwick which gives a solution to the comparison problem in higher category theory. We give a brief axiomatization of the theory of (infty,n)-categories (and other closely related theories). From this we show that the space of homotopy theories satisfying these axioms is B(Z/2)^n, and hence any two theories satisfying the axioms are equivalent with very little ambiguity in _how_ they are equivalent. Examples of popular theories which satisfy these axioms will be provided along with a spattering of applications.   

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