Formulation
For a symmetric monoidal -category which is fully dualizable and every -morphism of which is adjointable, for , there is a bijection between the -valued symmetric monoidal functors of the cobordism category and the objects of .
In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, [1] concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the details of his approach have yet to appear in the literature as of 2022. [2] [3] [4] In 2021, Daniel Grady and Dmitri Pavlov claimed a complete proof of the cobordism hypothesis, as well as a generalization to bordisms with arbitrary geometric structures. [4]
For a symmetric monoidal -category which is fully dualizable and every -morphism of which is adjointable, for , there is a bijection between the -valued symmetric monoidal functors of the cobordism category and the objects of .
Symmetric monoidal functors from the cobordism category correspond to topological quantum field theories. The cobordism hypothesis for topological quantum field theories is the analogue of the Eilenberg–Steenrod axioms for homology theories. The Eilenberg–Steenrod axioms state that a homology theory is uniquely determined by its value for the point, so analogously what the cobordism hypothesis states is that a topological quantum field theory is uniquely determined by its value for the point. In other words, the bijection between -valued symmetric monoidal functors and the objects of is uniquely defined by its value for the point.
In mathematics and computer science, currying is the technique of converting a function that takes multiple arguments into a sequence of functions that each takes a single argument. For example, currying a function that takes three arguments creates three functions:
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In mathematics, a monoidal category is a category equipped with a bifunctor
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In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. This means that, given a cohomology theory
,
In mathematics, especially in category theory, a closed monoidal category is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.
In mathematics, a commutativity constraint on a monoidal category is a choice of isomorphism for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have for all pairs of objects .
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory, . Jean Dieudonné used this to characterize Frobenius algebras. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.
This is a glossary of properties and concepts in category theory in mathematics.
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In mathematics, a highly structured ring spectrum or -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an -ring is called an -ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.
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In mathematics, specifically in category theory and algebraic topology, the Baez–Dolan stabilization hypothesis, proposed in, states that suspension of a weak n-category has no more essential effect after n + 2 times. Precisely, it states that the suspension functor is an equivalence for .
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