7 - Weighted limits and colimits

Summary

“Weighted” notions of limits and colimits are necessary to describe a comprehensive theory of limits and colimits in enriched categories. Weighted limits extend classical limits in two ways. First, the universal property of the representing object is enriched: an isomorphism of sets is replaced by an isomorphism in the base category for the enrichment. Second, and most interestingly, the shapes of cones that limits and colimits represent are vastly generalized. To build intuition, we begin our journey by examining the role played by weighted limits and colimits in unenriched category theory. In the unenriched context, weighted limits reduce to classical ones. Despite this fact, this perspective can be conceptually clarifying, as we see in the examples presented in this chapter and in Chapter 14.

After first gaining familiarity in the unenriched case, we define internal hom-objects for ν-Cat, which will be necessary to encode the enriched universal properties of weighted limits and colimits. We then introduce the general theory of weighted limits and colimits in a ν-category, also called ν-limits and ν-colimits; describe an important special case; and discuss ν-completeness and ν-cocompleteness. We close this chapter with applications to homotopy limits and colimits, establishing their previously advertised local universal property.

Weighted limits in unenriched category theory

The limit of a diagram F: C → M is an object of M that represents the functor M^op → Set that maps m ∈ M to the set of cones over F with summit m.