文档介绍：General abstract nonsense:
a brief introduction to category theory
Brad Hannigan-Daley
University of Waterloo
Motivation?
Let V be a finite-dimensional vector space over a field K. The dual of V is the set of linear functionals
V∗= {φ: V → K : φ is K-linear},
which we equip with pointwise addition and scalar multiplication, to turn it into a vector space over K. Let
∗ K ∗
us choose a basis {b1, . . . , bn} for V. For 1 ≤ i ≤ n, we define a linear map bi : V → by setting bi (bj) to
∗∗
be 1 if i = j and 0 otherwise, then extending this function linearly. It is an easy exercise that {b1, . . . , bn}
is a basis for V∗, and this correspondence gives an isomorphism V =∼ V∗. We can then immediately
conclude also that V is isomorphic to its double-dual V∗∗= (V∗)∗. However, there is an important
difference between these isomorphisms. Observe that our isomorphism V =∼ V∗ depends on our choice of
basis {bi}, and is in this sense not a “natural” isomorphism in the sense of being independent of any
choices we make. On the other hand, there is a “natural” isomorphism from V to V∗∗ given by assigning
∗∗∗
each x ∈ V to the functional σx ∈ V given by σx : V → K : φ 7→φ(x). (That this is an isomorphism is
left as exercise.) Category theory gives us a framework within which we can clarify and generalize, among
other things, this vague intuition of “natural isomorphism”