nLab
continuous functor

Definition

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

$$F(\lim A)\simeq \lim (F\circ A).$$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 $C$ 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 $F:D^{\mathrm{op}}\to C$ a diagram and $c\in C$, the covariant hom-functor ${\mathrm{Hom}}_C(c,-):C\to \mathrm{Set}$ satisfies by definition of limit

  $${\mathrm{Hom}}_C(c,\lim F)\simeq \lim \mathrm{Hom}(c,F(-)).$$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