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analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism topos, homotopy topos type theory, homotopy type theory …

nLab CW approximation Contents Context Homotopy theory homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Definitions Paths and cylinders Homotopy groups Basic facts Theorems Topology topology (point-s…

model category, model -category Definitions Morphisms Universal constructions Refinements Producing new model structures Presentation of -categories Model structures for -groupoids for equivariant -groupoids fine model structure on topological G-spaces coarse model structure on topological G-spaces (Borel model structure) for rational -groupoids for rational equivariant -groupoids for -groups for…

Special and general types Special notions Variants Extra structure Operations Theorems algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on…

nLab String Theory Contents Context String theory Ingredients Critical string models Extended objects Topological strings Backgrounds Phenomenology Physics This page is to record the reference: on string theory, first bosonic string theory, then superstring theory, with an outlook on M-theory. See also: Contents Volume 1 An introduction to the bosonic string 1 A first look at s…

fields and particles in particle physics and in the standard model of particle physics: photon - electromagnetic field (abelian Yang-Mills field) W, Z, B-boson - electroweak field (Yang-Mills field) gluon - strong nuclear force (Yang-Mills field) graviton - field of gravity infraparticle matter field fermions (spinors, Dirac fields) (also: antiparticles) hadrons (bound states of the above quarks)…

On boundary conformal field theory: The graded pivotal bicategory of B-twisted affine LG models is studied in detail in Orbifolds of defects are studied in Ilka Brunner, Daniel Roggenkamp, Defects and Bulk Perturbations of Boundary Landau-Ginzburg Orbifolds, JHEP 0804 (2008) 001, (arXiv:0712.0188) Nils Carqueville, Ingo Runkel, Orbifold completion of defect bicategories, (arXiv:1210.6363) Ilka Br…

Relation to D=3 N=2 super Yang-Mills theory: On physics interpretation of quantum elliptic cohomology:

Eric R. Sharpe Eric Sharpe works on theoretical physics related to string theory with an emphasis on structural aspects and mathematical formalization. He got his Phd under Edward Witten. Sharpe has early on realized the role of higher differential geometry in the form of stacks and gerbes in string theory. He identified string theorists’s “discrete torsion” with the equivariant cohomology of bun…

There is a little site notion of the Zariski topology, and a big site notion. As for the little site notion: the Zariski topology on the set of prime ideals of a commutative ring is the smallest topology that contains, as open sets, sets of the form where ranges over elements of . As for the big site notion, the Zariski topology is a coverage on the opposite category CRing of commutative rings. T…

On categorical semantics for Martin-Loef type theory: On type universes in homotopy type theory: On the univalence axiom:

representation, 2-representation, ∞-representation group, ∞-group group algebra, algebraic group, Lie algebra vector space, n-vector space affine space, symplectic vector space action, ∞-action module, equivariant object bimodule, Morita equivalence induced representation, Frobenius reciprocity Hilbert space, Banach space, Fourier transform, functional analysis orbit, coadjoint orbit, Killing for…

Special and general types Special notions Variants Extra structure Operations Theorems Twisted de Rham cohomology is the twisted cohomology-version of de Rham cohomology, a simple example of twisted differential cohomology. For degree-3 twists this is the codomain of the twisted Chern character on twisted K-theory, and in its orbifold cohomology-generalization it is the codomain of the twisted eq…

equality (definitional, propositional, computational, judgemental, extensional?, intensional?, decidable) identity type, equivalence of types, definitional isomorphism isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category natural equivalence, natural isomorphism gauge equivalence Examples. fiber product, pullback homotopy pullback The par…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … There are many possible definitions of the (two-sided, Dedekind) real numbers type in dependent type theory. If the dependent type theory has a type universe or a type of all proposit…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism topos, homotopy topos type theory, homotopy type theory …

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes function algebras on ∞-stacks Constructions loop space object, free loop space object fundamental ∞-groupoid in a locally ∞-connected (…

Special and general types Special notions Variants Extra structure Operations Theorems algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on…

target space, background gauge field twisted smooth cohomology in string theory landscape of string theory vacua Formalism Definition Spacetime configurations Properties Spacetimes Quantum theory

Moss E. Sweedler is professor emeritus at Cornell. He has pioneered Hopf algebra theory. His work on noncommutative extensions dealing with Sweedler corings and related work of his colleague Chase from Cornell on noncommutative Galois theory, predates Hopf-Galois theory from the 1970s. Moss E. Sweedler: Hopf algebras, Mathematics Lecture Note Series, Benjamin (1969) [ISBN:9780805392548] M. E. Swe…