Limit Preservation from Naturality

A functor G : C → D is said to preserve limits of a diagram D : I → C if it sends any limiting cone from x to D to a limiting cone from G(x) to G◦D. When G preserves limits of a diagram D this entails directly that there is an isomorphism G(lim ←−ID) ∼= lim ←−I(G ◦D) between objects. In general, such an isomorphism alone is not sufficient to ensure that G preserves limits. This paper shows how, with minor side conditions, the existence of an isomorphism natural in the diagram D does ensure that limits are preserved. In particular, naturality in the diagram alone is sufficient to yield the preservation of connected limits. At the other extreme, once terminal objects are preserved, naturality in the diagram is sufficient to give the preservation of products. General limits, which factor into a product of connected limits, are treated by combining these results. In particular, it is shown that a functor G : C → D between complete categories is continuous if there is an isomorphism G(lim ←−ID) ∼= lim ←−I(G ◦ D) natural in D ∈ [I, C], for any small category I. It is indicated how a little calculus of ends, in which the judgements are natural isomorphisms between functors, is useful in establishing continuity properties of functors.