# 40. Adjoint functors

In this lecture we first define *limits*** **in category theory (this is the dual notion to colimits, which we studied last semester in Lecture 16.)

Next, we define *adjoint functors*** **and investigate some examples. We then prove what is arguably the only “non-trivial” theorem in category theory that we have seen in the entire course: that adjoint functors preserve (co)limits. (The Acyclic Models Theorem from Lecture 23** **is another non-trivial abstract theorem, but this was not really a purely category-theoretic result.)

For us, the main use of adjoint pairs will come next week. We will construct two functors on the pointed homotopy category $\mathsf{hTop}_*$ (namely, the *loop space*** **functor and the *reduced suspension*** **functor), and prove that they form an adjoint pair.

Comments and questions?