Universals and Limits
Universal constructions appear throughout mathematics in various guises — as universal arrows to a given functor, as universal arrows from a given functor, or as universal elements of a set-valued functor. Each universal determines a representation of a corresponding set-valued functor as a hom-functor. Such representations, in turn, are analyzed by the Yoneda Lemma. Limits are an important example of universals — both the inverse limits ( = projective limits = limits = left roots) and their duals, the direct limits ( = inductive limits = colimits = right roots). In this chapter we define universals and limits and examine a few basic types of limits (products, pullbacks, and equalizers ...). Deeper properties will appear in Chapter IX on special limits, while the relation to adjoints will be treated in Chapter V.
KeywordsNatural Transformation Natural Isomorphism Finite Product Forgetful Functor Contravariant Functor
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- The Yoneda Lemma made an early appearance in the work of the Japanese pioneer N. Yoneda (private communication to Mac Lane) ; with time, its importance has grown.Google Scholar
- Representable functors probably first appeared in topology in the form of “universal examples”, such as the universal examples of cohomology operations (for instance, in J. P. Serre’s 1953 calculations of the cohomology, modulo 2, of Eilenberg-Mac Lane spaces).Google Scholar
- Universal arrows are unique only up to isomorphism; perhaps this lack of absolute uniqueness is why the notion was slow to develop. Examples had long been present; the bold step of really formulating the general notion of a universal arrow was taken by Samuel in 1948; the general notion was then lavishly popularized by Bourbaki. The idea that the ordinary cartesian products could be described by universal properties of their projections was formulated about the same time (Mac Lane [1948, 1950]). On the other hand the notions of limit and colimit have a long history in various concrete examples. Thus colimits were used in the proofs of theorems in which infinite abelian groups are represented as unions of their finitely generated subgroups. Limits (over ordered sets) appear in the p-adic•numbers of Hensel and in the construction of Cech homology and cohomology by limit processes as formalized by Pontrjagin. An adequate treatment of the natural isomorphisms occurring for such limits was a major motivation of the first Eilenberg-Mac Lane paper on category theory . E. H. Moore’s general analysis (about 1913) used limits over certain directed sets. In all these classical cases, limits appeared only for functors F: J—+C with J a linearly or partly ordered set. Then Kan  took the step of considering limits for all functors, while Freyd [ 1964 ] for the general case used the word “root” in place of “limit”. His followers have chosen to extend the original word ‘limit“ to this general meaning. Properties special to limits over directed sets will be studied in Chapter I X.Google Scholar