group in nLab
For example, Eilenberg-MacLane spaces K ( G , n ) K(G,n) have the property that homotopy classes of maps into them are in bijection with cohomology groups H n ( ...
One of the exceptional Lie groups. 2. Definition. Consider the vector space. W ≔ ∧ 2 ...
Contents. 1. Idea; 2. Properties. Representations of maximal compact subalgebra; As U-duality group of 2d supergravity.
Introduction to Stable Homotopy Theory in nLab
A double loop space is a group with some commutativity structure (“Eckmann-Hilton argument”), a triple loop space has more commutativity ...
Given a topological group G G , a (continuous) G G -set is a set X X equipped with a continuous group action μ : G × X → X \mu: G \times X \to X ...
groupal model for universal principal infinity-bundles in nLab
For G G a model for an ∞-group, there is often a model for the universal principal ∞-bundle E G \mathbf{E}G that itself carries a group ...
This entry discusses line objects, their multiplicative groups and additive groups in generality. For the traditional notions see at affine line.
Mapping spaces into H-groups ... If K K is an H H -group then for any topological space X X , the set of homotopy classes [ X , K ] [X,K] has a ...
Notably when abelian groups are generalized to their analogs in stable homotopy theory, namely to spectra, the corresponding internal monoids ...
Examples of this phenomenon include the monad for monoidal categories, symmetric monoidal categories, braided monoidal categories, categories ...
Opposite of the opposite. The opposite of an opposite category is the original category: ( C op ) op = C . (C^{op})^{op} = C \,. This is also ...
Being an abelian group, every delooping n-groupoid B n ( ℤ / 10 ) \mathbf{B}^n (\mathbb{Z}/{10}) exists. Carrying is a 2-cocycle in the group ...
Structure groups of heaps · (i) the bijections t ( a , b , − ) t(a,b,-) and t ( a ′ , b ′ , − ) t(a',b',-) coincide · (ii) t ( a , b , b ′ ) = a ′ ...
As such, the group G 2 G_2 is a higher analog of the symplectic group (which is the group that preserves a canonical 2-form on any ℝ 2 n ...
A mostly unrelated notion from group theory is order of a group, meaning the cardinality | G | |G| of the underlying set of a group G G , ...
For any finite group the number of its conjugacy classes is equal to the number of its irreducible representations. For finite groups of Lie ...
The collection of characters is itself an abelian group under the pointwise multiplication, this is called the character lattice Hom ( G , k × ) ...
By the above discussion of homotopy groups, it follows (by Chern-Weil theory) that the first invariant polynomials on the Lie algebra 𝔢 8 \ ...
This is the original notion of concrete group due to Évariste Galois. Abstracting from this, we get the modern notion of abstract group as a set ...
It is a theorem that a semigroup homomorphism between groups must be a monoid homomorphism (and additionally must preserve inverse elements, ...