Index

Monoids in tensor categories | Monoids in tensor categories

2-category: Monoids in tensor categories
2-functor: Monoids in tensor categories | Monoids in tensor categories
identity: Monoids in tensor categories
pasting: Monoids in tensor categories

2-functor

A-point

Hopf algebras

A-module

action: Monoids in tensor categories
arrow: Monoids in tensor categories
left: Monoids in tensor categories

action

Representations of quantum groups | Monoids in tensor categories | Tannaka duality

of an A-module: Monoids in tensor categories

adjoining

an antipode: Adjoining an antipode to
an inverse for the determinant: Hopf algebras
left-dual objects: Adjoining an antipode to

adjoint

left: Tannaka duality
left 2-adjoint: Tannaka duality
Lie algebra: Algebras

algebra

R-algebra: Algebras
exterior: Algebras
Hopf: Hopf algebras
morphism: Algebras
of endomorphisms: Algebras
of formal power series: Coalgebras and bialgebras
opposite: Algebras
over R: Algebras
polynomial R-algebra: Coalgebras and bialgebras
polynomial algebra: Coalgebras and bialgebras
symmetric: Coalgebras and bialgebras
tensor: Algebras
universal: Tensor functors and Yang-Baxter

Algebraic Geometry

algebraic varieties

antipode

Duality between geometry and | Hopf algebras | Monoids in tensor categories | Adjoining an antipode to

appropriate

representation: Tannaka duality

arrow

between monoid arrows: Monoids in tensor categories
composition: Internal homs and duals
monoid: Monoids in tensor categories
of category: Duality between geometry and

Associativity

Revision of basic structures | Revision of basic structures | Tensor categories

autonomous

left: Internal homs and duals
left/right: Internal homs and duals

balanced

YB-operator: Tensor functors and Yang-Baxter
bialgebra: Tensor categories

Banach spaces

example of closed symmetric tensor category: Internal homs and duals

bialgebra

R-bialgebra: Coalgebras and bialgebras
balanced: Tensor categories
braided: Tensor categories
commutative over k: Duality between geometry and
matrix bialgebra: Coalgebras and bialgebras
morphism: Coalgebras and bialgebras
quasitriangular: Tensor categories
symmetric: Tensor categories
triangular: Tensor categories

biclosed

bijection

canonical: Revision of basic structures

bilinear

R-bilinear: Modules and tensor products
universal bilinear function: Modules and tensor products

bimodule

left R-/right S-bimodule: Modules and tensor products

bimonoid

balanced: Monoids in tensor categories
cobalanced: Monoids in tensor categories
cobraided: Monoids in tensor categories
cotortile: Monoids in tensor categories
in V: Monoids in tensor categories
in: Monoids in tensor categories
strict quasi-bimonoid: Monoids in tensor categories
tortile: Monoids in tensor categories | Tannaka duality
with a braiding: Monoids in tensor categories
with a twist: Monoids in tensor categories

braid

category: Tensor categories
composition: Tensor categories
group, of Artin: Tensor categories
tensor product: Tensor categories

braided

bialgebra: Tensor categories
tensor category: Tensor categories

braiding

Tensor categories | Internal homs and duals | Monoids in tensor categories

element: Tensor categories
for a bimonoid: Monoids in tensor categories
for a tensor category: Tensor categories

C*-algebra

commutative: Duality between geometry and

cartesian product

of no sets: Revision of basic structures

category

of braids: Tensor categories
of monoids in V: Monoids in tensor categories
strict tensor: Monoids in tensor categories
tensor: Tensor categories

cauchy

H-comodule: Representations of quantum groups
module: Cauchy modules

closed

functor: Tensor functors and Yang-Baxter
left-closed: Internal homs and duals
tensor category: Internal homs and duals

coaction

coalgebra

morphism: Coalgebras and bialgebras
over a ring: Coalgebras and bialgebras
primitive element: Coalgebras and bialgebras
set-like element: Coalgebras and bialgebras

cobraiding

Dual coalgebras of algebras

cocauchy

cocommutative

coalgebra: Coalgebras and bialgebras

coend

coherence theorem

of MacLane: Tensor categories | Tannaka duality

commutative

Revision of basic structures | Algebras

R-algebra: Algebras
Lie algebra: Algebras
monoid: Revision of basic structures
rig: Revision of basic structures

Commutativity

commutator

defines a Lie bracket: Algebras

commute with diagonals

Representations of quantum groups | Representations of quantum groups

comodule

left: Representations of quantum groups

comodule morphism

Duality between geometry and | Duality between geometry and | Coalgebras and bialgebras

composite

pasted: Monoids in tensor categories

composition

arrow: Internal homs and duals
of braids: Tensor categories
vertical: Monoids in tensor categories

comultiplication

constraint

associativity: Tensor categories
left unit: Tensor categories
right unit: Tensor categories

convergent

convolution

product: Coalgebras and bialgebras
structure: Coalgebras and bialgebras

coordinate k-algebra

of general linear group: Duality between geometry and

coordinate algebra

quantum matrices: The quantum general linear

coprojection

coset

H-coset: Modules and tensor products

cotwist

counit

Duality between geometry and | Coalgebras and bialgebras | Internal homs and duals

derivation

diagonal

formal: Coalgebras and bialgebras
in category X: Duality between geometry and
ternary: Revision of basic structures

direct sum

of Lie algebras: Algebras
of modules: Cauchy modules

Distributive

Drinfeld V. G.

dual

left dual: Cauchy modules | Internal homs and duals
right dual: Internal homs and duals

dualizing object

left: Internal homs and duals
right: Internal homs and duals

Eli Cartan

end

Tannaka duality | Tannaka duality

is a limit: Tannaka duality

endomorphism algebra

enrich

abelian group with module structure: Modules and tensor products

equalizer

Monoids in tensor categories | Monoids in tensor categories

essentially

evaluation

functor: A tortile Yang-Baxter operator
morphism: Modules and tensor products

exponential series

exterior algebra

Fadeev L. D.

field

finitely generated

forgetful functor

Algebras | Monoids in tensor categories | Monoids in tensor categories | Tannaka duality

formal

diagonal: Coalgebras and bialgebras
power series: Coalgebras and bialgebras

Formal Tannaka Duality

free

constructions: Algebras
module: Modules and tensor products

free module

from R to S, generated by X: Modules and tensor products
functor: Tensor functors and Yang-Baxter

function

complex-valued: Duality between geometry and
continuous: Duality between geometry and
diagonal: Revision of basic structures
identity: Revision of basic structures

functor

essentially weak tensor: Monoids in tensor categories
evaluation: A tortile Yang-Baxter operator
forgetful: Monoids in tensor categories
self-adjoint: Hopf algebras
tensor: Tensor functors and Yang-Baxter
weak tensor functor: Monoids in tensor categories

funny superscripts

Gelfand duality

general linear group

commutative Hopf algebra: Duality between geometry and
coordinate k-algebra: Duality between geometry and

generate

generic point

geometric series

Revision of basic structures | Duality between geometry and

Grassmannian

algebra: The quantum general linear

group

R-algebra: Algebras
affine over k: Duality between geometry and
diagrammatic definition: Duality between geometry and
Lie group: Duality between geometry and
topological group: Duality between geometry and

homothety

A tortile Yang-Baxter operator

Hopf algebra

Duality between geometry and | Hopf algebras

commutative: Duality between geometry and

Hopf monoid

quasi: Monoids in tensor categories

Hurwitz polynomials

ideal

in an algebra: Algebras

identity

Revision of basic structures | Revision of basic structures | Revision of basic structures | Internal homs and duals

2-cell: Monoids in tensor categories

indeterminate

injective

morphism: Cauchy modules

internal hom

Internal homs and duals | Tannaka duality | Tannaka duality

left: Internal homs and duals
right: Internal homs and duals

Invertibility

invertible

isomorphism

Tensor functors and Yang-Baxter | Adjoining an antipode to

Joyal A.

k-algebra

coordinate algebra: Duality between geometry and
morphism: Revision of basic structures

Kobyzev, Yu

Kronecker delta

Hopf algebras

left 2-adjoint

left R-linear

left adjoint

left dual

as a functor: Internal homs and duals
of a module: Cauchy modules
of a signed set: Internal homs and duals

left-closed

Leibniz rule

Lie algebra

adjoint: Algebras
commutative: Algebras
direct sum: Algebras
Lie bracket: Algebras
morphism: Algebras
universal enveloping: Algebras

Lie bracket

linearly independent

The quantum general linear | Representations of quantum groups

MacLane

coherence theorem: Tensor categories | Tannaka duality

MacLane S.

Tannaka duality | Tannaka duality

Manin Y. I.

module

cauchy: Cauchy modules
finitely generated: Modules and tensor products
from R to S: Modules and tensor products
left R-module: Modules and tensor products
morphism: Modules and tensor products
projective: Cauchy modules
right R-module: Modules and tensor products

monoid

Revision of basic structures | Duality between geometry and | Monoids in tensor categories

morphisms preserve invertibility: Revision of basic structures
R-algebra: Algebras
affine over k: Duality between geometry and
arrow: Monoids in tensor categories
category: Monoids in tensor categories
commutative: Revision of basic structures
diagrammatic definition: Duality between geometry and
homomorphism: Revision of basic structures
morphism: Revision of basic structures
quasi-Hopf: Monoids in tensor categories

Morita theory

fundamental theorem: Cauchy modules

morphism

algebra: Duality between geometry and
comodule: Representations of quantum groups
evaluation: Modules and tensor products
map of varieties: Duality between geometry and
module morphism: Modules and tensor products
of C*-algebras: Duality between geometry and
of k-algebras: Revision of basic structures | Duality between geometry and
of R-algebras: Algebras
of bialgebras: Coalgebras and bialgebras
of coalgebras: Coalgebras and bialgebras
of Lie algebras: Algebras
of monoids: Revision of basic structures
of rigs: Revision of basic structures
retraction: Cauchy modules

multilinear

function: Modules and tensor products

multiplication

opposite: Modules and tensor products
scalar: Modules and tensor products

multiplicative matrices

natural family

Tensor categories

natural numbers

Monoids in tensor categories | Monoids in tensor categories

example of a rig: Revision of basic structures

natural transformation

non-commutating

indeterminates: The quantum general linear

object

of category: Duality between geometry and
terminal: Duality between geometry and
unit: Tensor categories

opposite

algebra: Algebras
multiplication: Modules and tensor products

pasted composite

pasting

2-cells: Monoids in tensor categories

Planck constant

Poincaré-Birkhoff-Witt

Duality between geometry and | Duality between geometry and

point

B-point: Duality between geometry and
B-point of a k-algebra: Duality between geometry and
as algebra morphism: Duality between geometry and
of an k-algebra: Duality between geometry and

primitive element

in a coalgebra: Coalgebras and bialgebras

product

in category X: Duality between geometry and
of modules: Cauchy modules
tensor product: Duality between geometry and

projection

projective

QIST

quadratic algebra

category: Internal homs and duals
morphism: Internal homs and duals

quantization

deforming commutative algebras to non-commutative ones: The quantum general linear

quantum

deformation: Hopf algebras
determinant: Hopf algebras
general linear group: The quantum general linear | Hopf algebras
group: Monoids in tensor categories | Tannaka duality
group over R: Monoids in tensor categories
inverse scattering transform: The quantum general linear
matrices: The quantum general linear
plane: The quantum general linear | Internal homs and duals
spaces: The quantum general linear
special linear group: The quantum general linear
superplane: The quantum general linear | Internal homs and duals

quantum group

cotortile bimonoid in: Monoids in tensor categories

quantum spaces

correspond to k-algebras: The quantum general linear

quasi-bimonoid

in V: Monoids in tensor categories

quasitriangular

bialgebra: Tensor categories

R-algebra

commutative: Algebras | Algebras
group: Algebras
monoid: Algebras
skew commutative: Algebras
symmetric: Algebras

R-coalgebra

representation

Representations of quantum groups | Monoids in tensor categories

appropriate: Tannaka duality
of group on monoid: Algebras

restriction of scalars

retract

retraction

morphism: Cauchy modules

reverse-arrow universal property

ribbons

YB-operator: Tensor functors and Yang-Baxter
tangles: Internal homs and duals

rig

commutative: Revision of basic structures
morphism: Revision of basic structures
natural numbers: Revision of basic structures

ring

with opposite multiplication: Modules and tensor products

Rivano N. S.

R-Lie algebra

Internal homs and duals | Tensor functors and Yang-Baxter

R-module

derivation: Algebras
two-sided: Algebras

scalar

multiplication: Revision of basic structures

self-adjoint

Hopf algebras

set-like element

in a coalgebra: Coalgebras and bialgebras

Shum M. C.

signed sets

skew commutative

small sets

Sophus Lie

source

of a tangle: Internal homs and duals | Internal homs and duals

space

seen from the other side of your brain: Duality between geometry and

span

strict

submodule

generated by a subset: Modules and tensor products

supergeometry

Revision of basic structures | Tensor categories

switch

symmetric

R-algebra: Algebras
tensor category: Tensor categories | Internal homs and duals

symmetry

for a tensor category: Tensor categories

taking off your belt

tangle

Tensor categories | Monoids in tensor categories

autonomous braided category: Internal homs and duals
geometric: Internal homs and duals
source: Internal homs and duals
tangles on ribbons: Internal homs and duals
tangles on strings: Internal homs and duals
target: Internal homs and duals

Tannaka

duality: Tannaka duality | Adjoining an antipode to
duality theorem: Tannaka duality

target

of a tangle: Internal homs and duals | Internal homs and duals

tensor

algebra: Algebras
functor: Tensor functors and Yang-Baxter
object: Monoids in tensor categories

tensor category

autonomous: Internal homs and duals
balanced: Tensor categories
braided: Tensor categories
closed: Internal homs and duals
free autonomous: Adjoining an antipode to
left-autonomous: Adjoining an antipode to
opposite: Tensor categories
strict: Tensor categories | Tensor categories | Monoids in tensor categories
symmetric: Tensor categories
tortile: Internal homs and duals | Internal homs and duals

tensor functor

balanced: Tensor functors and Yang-Baxter | Tannaka duality | Tannaka duality
braided: Tensor functors and Yang-Baxter | Tannaka duality
closed: Tensor functors and Yang-Baxter
left-closed: Tensor functors and Yang-Baxter
preserves dualizability: Tensor functors and Yang-Baxter
preserves duals: Tensor functors and Yang-Baxter
preserves products: Tensor functors and Yang-Baxter
right closed: Tensor functors and Yang-Baxter
strict: Tensor functors and Yang-Baxter
symmetric: Tensor functors and Yang-Baxter
takes YB-operator into YB-operator: Tensor functors and Yang-Baxter
weak: Tensor functors and Yang-Baxter

tensor product

Tensor categories

as composition of modules: Modules and tensor products
multiple: Modules and tensor products
of R-modules: Algebras
of braids: Tensor categories
represent bilinear function as module morphism: Modules and tensor products

tensor-hom

terminal object

twist

Tensor categories | Internal homs and duals | Monoids in tensor categories

element: Tensor categories

two-sided

R-module: Algebras

Ulbrich K.-H.

Tannaka duality | Tannaka duality

unit

left: Tensor categories
object: Tensor categories
right: Tensor categories

universal

algebra: Tensor functors and Yang-Baxter

universal enveloping algebra

Algebras | Coalgebras and bialgebras | Tensor functors and Yang-Baxter

is a cocommutative bialgebra: Coalgebras and bialgebras

universal property

end: Tannaka duality
for internal hom: Internal homs and duals
internal hom: Tannaka duality
reverse-arrow: Representations of quantum groups

up to coherent isomorphism

vector space

over R: Modules and tensor products

weak tensor functor

essentially: Monoids in tensor categories
takes monoids to monoids: Monoids in tensor categories

whisker

Monoids in tensor categories | Monoids in tensor categories | Monoids in tensor categories | Monoids in tensor categories | Monoids in tensor categories

Yang-Baxter