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Types are weak ωgroupoids
, 2008
"... We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids. ..."
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We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids.
Multitensor lifting and strictly unital higher category theory
"... Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories ..."
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Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories and the Crans tensor product of Gray categories as part of this framework. We define weak ncategories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak ncategories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)categories with strict units. 1.