pdf file

Examples 2.1.9.
1. It is possible to give a category whose objects are finitely many points and whose morphisms
are arrows.
2. The left modules over any algebra A form a category which we denote by A-mod. The
right A-modules form a category as well which we denote by mod-A. In general, these
categories are not related.
3. If A is a K-algebra, the morphisms Hom(V, W ) have more structure than the one of a
set: they are K-vector spaces, and the composition is K-bilinear. One says that such a
category is enriched over the category of K-vector spaces.
4. Consider a category with a single object • that is enriched over the category of K-vector
spaces. Then End(•) is a K-algebra.
5. A category in which all morphisms are isomorphisms is a called a groupoid. A groupoid
with single object • has a single Hom-space Hom(•, •) that is is a group.
An important example of a groupoid is the fundamental groupoid of a topological space
M: its objects are the points of the space M, a morphism from p ∈ M to q ∈ M is a
homotopy class of paths from p to q.
For the next observation, we need the following notion:
Proposition 2.1.10.
Let (A, μ A , η A ) and (B, μ B , η B ) be unital associative K-algebras. Then the tensor product A⊗B
has a natural structure of an associative unital algebra determined by
and η A⊗B := η A ⊗ η B .
(a ⊗ b) ∙ (a ′ ⊗ b ′ ) := aa ′ ⊗ b ∙ b ′ for all a, a ′ ∈ A, b, b ′ ∈ B
Put differently, the multiplication μ A⊗B is the map
A ⊗ B ⊗ A ⊗ B id A⊗τ⊗id
−→
B
μ
A ⊗ A ⊗ B ⊗ B
A ⊗μ
−→
B
A ⊗ B ,
with τ the twist map τ : a ⊗ b ↦→ b ⊗ a.
Observation 2.1.11.
Modules over a group algebra have more structure.
• Let V, W be K[G]-modules. Then the ground field K, the tensor product V ⊗ K W and the
dual vector space V ∗ := Hom K (V, K) can be turned into K[G]-modules as well by
g.1 := 1 for all g ∈ G
g.(v ⊗ w) := g.v ⊗ g.w for all g ∈ G, v ∈ V and w ∈ W
(g.φ)(v) := φ(g −1 .v) for all g ∈ G, v ∈ V and φ ∈ V ∗ .
(In physics, the representation on K is used to describe invariant states, and the representation
on V ⊗ W corresponds to “coupling systems” for symmetries leading to multiplicative
quantum numbers.)
10