Browsing by Author "Haderi, Redi"

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    2-fold structures and homotopy theory
    (2023-01) Haderi, Redi
    It is well-known that correspondences between categories, also known as profunctors, serve in classifying functors. More precisely, every functor F : X → A straightens into a lax mapping χF : A → Catprof from A into a 2-category of categories and profunctors ([45]). We give a conceptual treatment of this fact from the lens of double category theory, contending the latter to be most natural environment to express this result. Then we venture into the world of simplicial sets and prove an analogous theorem. The notion of correspondence is easy to extend to simplicial sets, but a suitable double category may not be formed due to the lack of a natural tensor product. Nonetheless, we show that there is a natural simplicial category structure once we invoke higher correspondences. In proving our result we extend some notions from double category theory into the world of simplicial categories. As an application we obtain a realization of Lurie’s prediction that inner fibrations are classified by mappings into a higher category of correspondences between ∞-categories.
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    A simplicial category for higher correspondences
    (Springer Science and Business Media B.V., 2022-12-27) Haderi, Redi
    In this work we propose a realization of Lurie’s prediction that inner fibrations p : X → A are classified by A-indexed diagrams in a “higher category” whose objects are ∞-categories, morphisms are correspondences between them and higher morphisms are higher correspondences.We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and ∞-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well.