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12
Higher dimensional algebra VII: Groupoidification
, 2010
"... Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector space ..."
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Cited by 9 (3 self)
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Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normalordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang–Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field Fq. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of Fq representations of a simplylaced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify—or more precisely, groupoidify—the positive part of the quantum group associated to the quiver.
A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES
"... Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “l ..."
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Cited by 4 (0 self)
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Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous
Super, Quantum and NonCommutative Species
, 2009
"... Dedicated to the memory of GianCarlo Rota. We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and noncommutative combinatorics. Via the usual dual ..."
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Cited by 3 (3 self)
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Dedicated to the memory of GianCarlo Rota. We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and noncommutative combinatorics. Via the usual duality between algebra and geometry, these constructions provide categorifications for various types of affine spaces, thus our works may be regarded as a starting point towards the construction of a categorical geometry.
Nonablian cocycles and their σmodel QFTs
, 2008
"... Nonabelian cohomology can be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to ∞groupoids or even to general ∞categories. Cocycles in nonabelian cohomology in particular represent higher princ ..."
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Nonabelian cohomology can be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to ∞groupoids or even to general ∞categories. Cocycles in nonabelian cohomology in particular represent higher principal bundles (gerbes) – possibly equivariant, possibly with connection – as well as the corresponding associated higher vector bundles. We propose, expanding on considerations in [13, 34, 5], a systematic ∞functorial formalization of the σmodel quantum field theory associated with a given nonabelian cocycle regarded as the background field for a brane coupled to it. We define propagation in these σmodel QFTs and recover central aspects of groupoidification [1, 2] of linear algebra. In a series of examples we show how this formalization reproduces familiar structures in σmodels with finite target spaces such as DijkgraafWitten theory and the Yetter model. The generalization to
Categorifying Fundamental Physics
"... Despite the incredible progress over most of the 20th century, and a continuing flow of new observational discoveries — neutrino oscillations, dark matter, dark energy, and evidence for inflation — theoretical research in fundamental physics seems to be in a ‘stuck ’ phase. So, now more than ever, i ..."
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Despite the incredible progress over most of the 20th century, and a continuing flow of new observational discoveries — neutrino oscillations, dark matter, dark energy, and evidence for inflation — theoretical research in fundamental physics seems to be in a ‘stuck ’ phase. So, now more than ever, it seems important to reexamine basic assumptions and seek fundamentally new ideas. Work along these lines is inherently risky: many different directions must be explored, since while few will lead to important new insights, it is hard to know in advance which these will be. For this reason, I have ceased for now to work on loop quantum gravity, and begun to rethink basic mathematical structures in physics. The unifying idea behind this multipronged project is ‘categorification’, or in simple terms: giving up the naive concept of equality. While equations play an utterly fundamental role in physics, and this is unlikely to change, equations between elements of a set often arise as a shorthand for something deeper: isomorphisms between objects in a category.
Nonabelian homotopical cohomology,
"... higher fiber bundles with connection, and their σmodel QFTs ..."
processes. Classical Sets
, 2010
"... Notes for a seminar in the IST TQFT Club about the BaezDolan groupoidification and its extensions, as applied to some toy models in ..."
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Notes for a seminar in the IST TQFT Club about the BaezDolan groupoidification and its extensions, as applied to some toy models in
Nonabelian homotopical cohomology,
"... higher fiber bundles with connection, and their σmodel QFTs ..."