nLab
continuous functor

Definition

A functor F:CDF : C \to D is continuous if it preserves all small (weighted) limits that exist in CC, i.e. if for every small category diagram A:ECA : E \to C in CC there is an isomorphism

F(limA)lim(FA). F(\lim A) \simeq \lim (F\circ A) \,.

Since all limits can be obtained from (small) products and binary equalizers, it follows that a functor is continuous if and only if it preserves all products and all binary equalizers.

Relation to other concepts

  • The adjoint functor theorem states that any continuous functor between complete categories has a left adjoint if it satisfies a certain ‘small solution set’ criterion.

  • If CC has finite limits, then functors commuting with these finite limits are precisely what are called left exact functors. Sometimes they are called “finitely continuous.”

Examples

  • The archetypical example is the Set-valued hom-functor: its continuity in both arguments is indeed equivalent to the very definition of limit: for F:D opCF : D^{op} \to C a diagram and cCc \in C, the covariant hom-functor Hom C(c,):CSetHom_C(c,-) : C \to Set satisfies by definition of limit
    Hom C(c,limF)limHom(c,F()). Hom_C(c, \lim F) \simeq \lim Hom(c,F(-)) \,.

Warnings

  1. Topologists sometimes use “continuous functor” to mean a functor enriched over Top, since a functor between topologically enriched categories is enriched iff its actions on hom-spaces are continuous functions.

  2. Sheaf-theorists sometimes say “continuous functor” for a cover-preserving functor between sites, with the intuition being that it generalizes the inverse image induced by a continuous function of topological spaces.

  3. H. Bass in his treatment of K-theory uses the older term ‘right continuous functor’ for the dual notion of cocontinuous functor in a version which is additive. If the domain of an additive functor which commutes with direct sums is a cocomplete category, then the functor automatically has right adjoint. Following this fact, some people in ring theory and noncommutative geometry use the simple term ‘continuous functor’ for a functor with a right adjoint (even if the domain abelian category is not cocomplete). In general, of course, this is just a bit more than cocontinuous in the standard sense.

Left I could understand, but right? —Toby

The way I rewrote it explains it. It is unfortunate that the Eilenberg-Watts theorem treated in Bass was using only right adjoint functors so later they dropped word right. – Zoran

Thanks. —Toby

Revised on November 7, 2010 18:12:22 by Mike Shulman (71.137.3.108)