6.2 Analytic functors
6.2.3 Analytic functors and finiteness
Joyal [1986] also gave yet another characterization of analytic functors, namely, those which preserve filtered colimits, cofiltered limits, and weak pullbacks. It is instructive to use this characterization as a lens to consider some examples of functors which are
not analytic.
Definition 6.2.2. A filtered category C [Ad´amek et al., 2002] is one which “has all finite cocones”, that is, for any finite collection of objects and morphisms in C, there is some object C ∈ C with morphisms from all the objects in the collection to C, such that all the relevant triangles commute.
Equivalently, and more simply, a filtered category is one for which
• there exists at least one object;
• any two objectsC1, C2 ∈Chave an “upper bound”, that is, an object C3 with
morphisms
C1 //C3 oo C2;
• and finally, any two parallel morphisms C1
f //
g //C2 also have an “upper bound”,
that is, another morphism
C1 f // g //C2 h // C3 such that f;h=g;h.
These binary upper bound operations on objects and morphisms may be used to inductively “build up” cocones for arbitrary diagrams in C.
This can be seen as a “categorification” of the notion of adirected set (also known
as a filtered set), a preorder in which any two elements have an upper bound. Cat-
egories can be seen as generalizations of preorders in which multiple morphisms are allowed between each pair of objects, so the above definition has to extend the idea of pairwise upper bounds to apply to parallel morphisms as well as objects; in a preorder there are no parallel morphisms so this does not come up.
Example. Any category with a terminal object is filtered: the terminal object may
be taken as the upper bound of any two objects, and the unique morphism to the terminal object as the upper bound of any two parallel morphisms.
Example. The poset (N,6), considered as a category whose objects are natural num-
bers, with morphisms m 6 n, is a filtered category. The upper bound of any two objects is their maximum, and there are no parallel morphisms to consider.
0 //1 //2 //3 //. . .
Note that filteredness only requires that every finite collection of objects have an upper bound; in particular, in this example it is not true of infinite collections of objects. For example, the set of all even numbers has no upper bound inN.
Example. Consider the category FinN⊆ whose objects are finite subsets of N and
whose morphisms are inclusion maps. That is, whenever S ⊆ T there is a single morphism ιST :S → T defined by ιST(s) = s. Since this is a nonempty preorder, to
see that FinN⊆ is filtered it suffices to note that any two finite sets S and T have
S∪T as an upper bound.
Example. Filtered categories can also be seen as a generalization of finitely cocomplete
categories, i.e. categories having all finite colimits. In particular, categories having all finite colimits can be characterized as those having an initial object, all binary coproducts, and all coequalizers: these are exactly parallel to the three criteria given above for filtered categories, with an extra “universal property” corresponding to each (for example, the binary coproduct of two objects is an upper bound along with a universal property).
Therefore, any (finitely) cocomplete category is automatically filtered: for exam- ple, Set, Grp, and Vec.
Recall that a diagram in C is a functor I → C from some “index category” I, which determines the “shape” of diagrams in C.
Definition 6.2.3. A filtered diagram in C is a functor I → C from a filtered index category I. A filtered colimit is a colimit of a filtered diagram.
That is, a filtered diagram in C is a diagram that “looks like” a filtered category “sitting inside” C. A filtered colimit is then just a normal colimit which happens to be taken over a filtered diagram.
Example. Let F :C→ C be an endofunctor on the category C. Suppose C contains
an initial object 0, and let ! denote the unique morphism 0 →C. Then consider the diagram
0 ! //F0 F! //F20 F2! //F30 //. . .
The colimit of this diagram is the least fixed pointµF, and is a filtered colimit since the diagram has the filtered poset (N,6) as its index category.
Example. Pushouts are an example of colimits which are not filtered, since pushouts
are colimits over a span X oo Z //Y , which is not filtered (X and Y do not necessarily have an upper bound).
Example. Recall the filtered poset FinN⊆ introduced earlier, consisting of finite sub-
sets of N and inclusion maps. The inclusion functor FinN⊆ ,→ Set allows viewing FinN⊆ as a diagram in Set, and we consider the (filtered) colimit of this diagram,
which must consist of some setS along with maps from all the finite subsets ofNinto
S, which commute with the inclusion maps among the finite subsets of N. In fact, it suffices to take N itself, together with the inclusion maps from each finite subset of N into N. Intuitively, N arises here as the disjoint union of all finite subsets of N, quotiented by the relationships induced by all the inclusion maps—which collapses the disjointness, resulting in a simple union of all finite subsets.
To see that this is universal, suppose we have a set X with mapsmS :S →X for
each finite S ⊂ N, such that the mS all commute with inclusion maps between the
finite subsets of N. Defineθ :N→X byθ(n) =m{n}(n). We must show that the mS
all factor through θ.
S mS ιS {k} o o m{k} N θ //X
Given some S ⊂ N and some k ∈ S, we have θ(ιS(k)) = θ(k) = m{k}(k); but this is
indeed equal tomS(k), since there exists an inclusion map {k} →S, and we assumed
the mS commute with inclusion maps.
Now consider the functor F := (−)N : Set → Set, which sends the set A to the
setAN of functions from
N toA [Trimble, 2014]. The claim is thatF is not analytic,
and in particular that it does not preserve the filtered colimit of FinN⊆, discussed
above. As we will see, the “problem” is that F corresponds to an infinite data type,
i.e. one which can contain infinitely many A values. In particular, F corresponds to the data type of infinite streams: a function N→A can be thought of as an infinite
stream ofA values, where the value of the function at n gives the value of A located at positionn in the stream.
We also consider how F acts on inclusion maps. The action of F on morphisms is given by postcomposition, so F sends the inclusion ι:S ,→T toι◦ −:SN →TN,
which is also an inclusion map: it sends the stream s : N → S to the stream ι◦s :
N → T, consisting of the application of ι to every element in s. That is, ι◦ − does
not actually modify any values of a stream, but simply codifies the observation that whenever S ⊆T, a stream containing only values from S may also be thought of as a stream containing only values from T (which simply happens not to include any values from T −S).
We saw above that the colimit of FinN⊆, considered as a diagram in Set, is N
(together with the obvious inclusion maps to N from each finite subset). F sends N to NN, the type of infinite streams of natural numbers. F also sends each inclusion
map S ,→ N to the inclusion SN ,→
NN, which allows a stream of S values to be
“upgraded” to a stream of natural numbers.
Now consider where F sends the diagram FinN⊆. F sends each finite set S ⊂ N
to the set of infinite streams of S values, SN, and it sends each inclusion S ,→ T to
the inclusion SN ,→TN. However, the colimit of this new diagram F(FinN
⊆) is not
NN, the set of streams of natural numbers, but instead the set of finitely supported
streams of natural numbers, that is, the set of all streams which contain only finitely many distinct elements. ThusF (colimFinN⊆)colim(F FinN⊆), and we conclude
that F is not analytic since it does not preserve filtered colimits.
Another example3 is given by the covariant power set functor P : Set → Set, which sends each set A to its power set P(A), the set of all subsets of A, and sends each functionf :A→B to the function P(f) :P(A)→P(B) which gives the image of a subset of A underf. P(N) is the set of all (finite and infinite) subsets of N, but colimP(FinN⊆) is the set of allfinite subsets ofN. Note, however, that the covariant
finite powerset functor F P :Set →Set, which sends each set A to the set of all its
finite subsets, is analytic; it corresponds to the species E·E.