In this lecture we start a block of the course in which we study the representation theory of quantum groups and its applications. We will see that they are intimately connected with braids and knots.

We start with some abstract definitions of monoidal and braided categories. A category C for our purposes is just

1. A collection of objects V, W, Z, U, ….

2. A specification of a set Mor(V, W) of morphisms for each V, W.

The sets Mor(V, W), Mor(Z, U) are disjoint unless V = Z and W = U.

3. A composition operation ∘ : Mor(W, Z) × Mor(V, W) → Mor(V, Z) with properties analogous to the composition of maps (such as associativity of ∘ where defined).

4. Every set Mor(V, V) should contain an identity element idV such that φ ∘ id = φ, id ∘ φ = φ for any morphism for which ∘ is defined.

A more formal treatment is in Mac Lane's book for anyone interested. In our case all objects will be concrete sets with structure (actually vector spaces equipped with linear maps of various kinds), all morphisms will be linear maps obeying various restrictions, and all categories will be equivalent to essentially small ones (i.e. we will not digress on topos theory and other subtleties). We are primarily going to use the language of category theory to keep our thinking clear. In particular, we indicate objects as V ∈ C by an abuse of set theory notations.