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.