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Space-Valued Diagrams, Type-Theoretically (Extended Abstract).
Nicolai Kraus,Christian Sattler +1 more
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It is shown how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory.Abstract:
Topologists are sometimes interested in space-valued diagrams over a given index category, but it is tricky to say what such a diagram even is if we look for a notion that is stable under equivalence. The same happens in (homotopy) type theory, where it is known only for special cases how one can define a type of type-valued diagrams over a given index category. We offer several constructions. We first show how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory. Further, we present a construction of diagrams over certain Reedy categories. As an application, we add the degeneracies to the well-known construction of semisimplicial types, yielding a construction of simplicial types up to any given finite level. The current paper is only an extended abstract, and a full version is to follow. In the full paper, we will show that the different notions of diagrams are equivalent to each other and to the known notion of Reedy fibrant diagrams whenever the statement makes sense. In the current paper, we only sketch some core ideas of the proofs.read more
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Two-Level Type Theory and Applications.
TL;DR: 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT, and a collection of tools are set up with the goal of making 2LTT a convenient language for future developments.
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Univalent higher categories via complete Semi-Segal types
Paolo Capriotti,Nicolai Kraus +1 more
TL;DR: The notion of a complete semi-Segal n-type can be taken as the definition of a univalent n-category as discussed by the authors, which is a simplicial approach to higher category theory.
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Homotopical inverse diagrams in categories with attributes
TL;DR: Kapulkin and Lumsdaine as mentioned in this paper define and develop the infrastructure of homotopical inverse diagrams in categories with attributes, and apply the present results to construct semi-model structures on categories of contextual categories.
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Homotopical inverse diagrams in categories with attributes
TL;DR: In this paper, the infrastructure of homotopical inverse diagrams in categories with attributes is defined and developed, which can be used to construct semi-model structures on categories of contextual categories.
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Internal $\infty$-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT
TL;DR: A notion of ∞-categories with families (∞-CwF’s) is developed, with a new construction of identity substitutions that allow for both univalent and non-univalent variations.
References
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Homotopy Invariant Algebraic Structures on Topological Spaces
John. M. Boardman,Rainer M. Vogt +1 more
TL;DR: In this article, the bar construction for topological-algebraic theories is discussed. But the authors focus on topological algebraic theories and do not consider topological algebras.
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A model for the homotopy theory of homotopy theory
TL;DR: In this article, the objects of a category may be viewed as models for homotopy theories, and it is shown that the category of such models has a well-behaved internal homobject.
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Abstract homotopy theory and generalized sheaf cohomology
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Univalence for inverse diagrams and homotopy canonicity
TL;DR: In this paper, a homotopical version of the relational and gluing models of type theory is described, and generalization to inverse diagrams and oplax limits is proposed.