6 - Adjoints, Limits and Kan Extensions

Summary

We introduce the fundamental notions of adjoint functor, limit, colimit and Kan extension for ∞-categories. We do this in a way which looks like the classical presentation with ordinary categories. In particular, we study the functorial properties of these notions. We study the existence of extensions of functors by small colimits as well as of the right adjoints of the latter. This is used to prove that the ∞-category of ∞-groupoids has both small limits and small colimits. We then construct and compute Kan extensions in contexts where we have enough small (co)limits. We study in particular pull-backs in ∞-categories, and show that they behave exactly as their classical counterparts. Finally, we prove a duality result: an equivalence of ∞-categories which expresses the duality between a small ∞-category and its opposite in terms of ∞-categories of small colimit preserving functors.