functor category

Functor categories


Given categories CC and DD, the functor category – written D CD^C or [C,D][C,D] – is the category whose


Functor categories serve as the hom-categories in the strict 2-category Cat.

In the context of enriched category theory the functor category is generalized to the enriched functor category.

In the absence of the axiom of choice (including many internal situations), the appropriate notion to use is often instead the anafunctor category.


Limits and colimits and closure

If DD has limits or colimits of a certain shape, then so does [C,D][C,D] and they are computed pointwise. (However, if DD is not complete, then other limits in [C,D][C,D] can exist “by accident” without being pointwise.)

If CC is small and DD is cartesian closed and complete, then [C,D][C,D] is cartesian closed. See at cartesian closed category for a proof.

Accessibility and local presentability

Functor categories enjoy the following accessibility and local presentability properties, as explained by Zhen Lin Low at nForum.

  • κ\kappa-accessible functors from a κ\kappa-accessible category to any accessible category form an accessible category. (It is not so easy to say what the accessibility rank is here.)

  • κ\kappa-accessible functors from a κ\kappa-accessible category to any locally λ\lambda-presentable category form a locally λ\lambda-presentable category.

  • Cocontinuous functors between locally presentable categories form a locally presentable category. More precisely, if CC and DD are locally κ\kappa-presentable, then so is [C,D][C,D].

  • Continuous accessible functors between locally presentable categories form the opposite of a locally presentable category. More precisely, if CC and DD are locally κ\kappa-presentable, then so is [C,D] rmop[C,D]^{\rm op}.

Indeed, the point is this: given a κ\kappa-accessible category 𝒞Ind κ(𝒜)\mathcal{C} \simeq Ind^\kappa (\mathcal{A}) (𝒜\mathcal{A} essentially small), the category of κ\kappa-accessible functors 𝒞𝒟\mathcal{C} \to \mathcal{D} (for arbitrary 𝒟\mathcal{D}; here by “κ\kappa-accessible” we mean simply “preserves κ\kappa-filtered colimits”) is naturally equivalent to the category of all 𝒜𝒟\mathcal{A} \to \mathcal{D}. It should be well known that:

  1. If 𝒟\mathcal{D} is accessible, then so is [𝒜,𝒟][\mathcal{A}, \mathcal{D}].

  2. If 𝒟\mathcal{D} is locally λ\lambda-presentable, then so is [𝒜,𝒟][\mathcal{A}, \mathcal{D}].

  3. Colimit-preserving functors out of a locally κ\kappa-presentable category are κ\kappa-accessible.

  4. A right adjoint between locally κ\kappa-presentable categories is κ\kappa-accessible if and only if its left adjoint is strongly κ\kappa-accessible (i.e. preserves κ\kappa-presentable objects as well as κ\kappa-filtered colimits); and every limit-preserving accessible functor between locally presentable categories is a right adjoint.

Statements 1 and 2 are proved in [Adamek and Rosický, Locally presentable and accessible categories], statement 3 is obvious, and statement 4 is a straightforward exercise. Thus the claims follow.

In general, accessible functors between accessible categories do not form an accessible category due to size issues. The best one can hope for is a class-accessible category. Let 𝒞\mathcal{C} be an accessible category that is not essentially small. Consider the category 𝒜\mathcal{A} of all accessible functors 𝒞Set\mathcal{C} \to \mathbf{Set}. This is the same as the smallest full replete subcategory of [𝒞,Set][\mathcal{C}, \mathbf{Set}] containing all representable functors and closed under small colimits. In particular, 𝒜\mathcal{A} is accessible if and only if 𝒜\mathcal{A} locally presentable. We claim 𝒜\mathcal{A} is not accessible.

Indeed, suppose 𝒜\mathcal{A} has a small generating family, say 𝒢\mathcal{G}. Then for some regular cardinal κ\kappa, every member of 𝒢\mathcal{G} is κ\kappa-accessible. So consider 𝒞(X,)\mathcal{C} (X, -) for some object XX that is not κ\kappa-presentable. (Such an XX exists because 𝒞\mathcal{C} is not essentially small.) Since 𝒢\mathcal{G} generates, there is a small diagram of κ\kappa-accessible functors whose colimit is 𝒞(X,)\mathcal{C} (X, -). But then 𝒞(X,)\mathcal{C} (X, -) is a retract of a κ\kappa-accessible functor and hence κ\kappa-accessible: a contradiction. That said, 𝒜\mathcal{A} is a class-locally presentable category.

Size issues

If CC and DD are small, then [C,D][C,D] is also small.

If CC is small and DD is locally small, then [C,D][C,D] is still locally small.

Even if CC and DD are locally small, if CC is not small, then [C,D][C,D] will usually not be locally small.

As a partial converse to the above, if CC and [C,Set][C,Set] are locally small, then CC must be essentially small; see Freyd & Street (1995).

Revised on August 7, 2017 04:42:06 by Joshua Hunt? (