nLab continuous functor

Definition

In ordinary (Set-based) category theory, a functor $F \colon C \to D$ is continuous if it preserves all small limits that exist in its domain $C$, i.e. given any small category diagram $A \colon E \to C$ in $C$ for which $\lim A$ exists, with universal cone $\lim A \to A$, application of $F$ to that cone results in a universal cone $F(\lim A) \to F \circ A$, thus making $F(\lim A)$ the limit of $F \circ A$.

Put slightly differently, the limit $\lim (F \circ A)$ exists and the natural map induced by the universal property of the limit

$$F(\lim A) \longrightarrow \lim (F\circ A)$$

is an isomorphism. In many and perhaps most cases, one refers to continuous functors $F \colon C \to D$ in cases where one already knows $C$ and $D$ are complete categories, i.e., where $\lim A$ and $\lim F \circ A$ exist as a matter of course.

In such cases, all limits can be obtained from (small) products and binary equalizers (see here), and so it follows that a functor from a complete category is continuous if and only if it preserves all products and all binary equalizers.

Examples

Example

The archetypical example is the Set-valued hom-functor: its continuity in both arguments (cf. hom-functors preserve limits) is indeed equivalent to the very definition of limit: for $F \colon D^{op} \to C$ a diagram and $c \in C$, the covariant hom-functor $Hom_C(c,-) : C \to Set$ satisfies by definition of limit

$$Hom_C(c, \lim F) \simeq \lim Hom(c,F(-)) \,.$$

Likewise for contravariant representable functors on $C$, i.e., functors of the form $C(-, c): C^{op} \to Set$, which take limits in $C^{op}$ to limits in $Set$, or colimits in $C$ to limits in $Set$.

Example

Every right adjoint functor is continuous, since right adjoints preserve limits.

In enriched category theory

One can also formulate limit preservation in terms of representable functors. This is especially appropriate in enriched category theory, where ordinary limits in terms of cones may no longer suffice and one should use weighted limits instead. Thus, working over a base of enrichment $V$, assumed to be a Bénabou cosmos for convenience, suppose given a ($V$-enriched) functor $F \colon C \to D$, a small weight $W\colon J \to V$, and a diagram $A \colon J \to C$ over that weight. Assume the weighted limit $\{W, A\}$ exists in $C$, i.e., suppose we have a representing object for the $V$-functor $c \mapsto V^{J}(W, C(c, A(-)))$, giving an isomorphism

$$C(c, \{W, A\}) \cong V^{J}(W, C(c, A(-)))$$

natural in $c$. In this situation, playing the role analogous to a universal cone is a universal $V$-natural transformation

$$W \to C(\{W, A\}, A(-))$$

and we say $F \colon C \to D$ preserves the weighted limit if the composite

$$W \to C(\{W, A\}, A(-)) \to D(F\{W, A\}, (F \circ A)(-))$$

induces, à la Yoneda, a $V$-natural isomorphism

$$D(d, F\{W, A\}) \cong V^J(W, D(d, (F \circ A)(-))).$$

Then we say that $F \colon C \to D$ is $V$-continuous if it preserves all weighted limits that exist in $C$.

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, or “finitely continuous” functors.

Warnings

1. It is not enough to demand that there exists an abstract isomorphism $F(\lim A) \cong \lim (F\circ A)$.

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

3. 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.

4. 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.

Related concepts

• cocontinuous functor

• complete category

• adjoint functor theorem

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

• Peter Freyd, Max Kelly, Categories of continuous functors, J. Pure. Appl. Algebra 2 (1972) 169-191 [doi:10.1016/0022-4049(72)90001-1]

• Max Kelly, Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]