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Idea Tannaka duality or Tannaka reconstruction theorems are statements of the form: if AA is a symmetry object (e.g. a locally compact topological group, Hopf algebra), represented on objects in a category DD, one may reconstruct AA from knowledge of the endomorphisms of the forgetful functor – the fiber functor – from the category Rep D(A)Rep_D(A) of representations of AA on objects of DD that re

Idea Type theory and certain kinds of category theory are closely related. By a syntax-semantics duality one may view type theory as a formal syntactic language or calculus for category theory, and conversely one may think of category theory as providing semantics for type theory. The flavor of category theory used depends on the flavor of type theory; this also extends to homotopy type theory and

In the archetypical case of the Cartesian monoidal category of finite sets this is Hotz’s notation (Hotz 65) for automata, while for finite-dimensional vector spaces with their usual tensor product this is Penrose’s notation (Penrose 71a, Penrose-Rindler 84) for tensor networks; but the same idea immediately applies more generally to any other monoidal category and yet more generally to bicategori

This is a wiki for collaborative work on Mathematics, Physics, and Philosophy — especially (but far from exclusively) from the higher structures point of view: with a sympathy towards the tools and perspectives of homotopy theory/algebraic topology, homotopy type theory, higher category theory and higher categorical algebra. Contents Purpose The nLab records and explores a wide range of mathematic