Examples of functors $\mathbf{Set} \to \mathbf{Set}$ which are not analytic

Question

Let $\mathbb{B}$ denote the groupoid of finite sets and bijections.

A functor $F : \mathbf{Set} \to \mathbf{Set}$ is analytic if it is the left Kan extension of some functor $G : \mathbb{B} \to \mathbf{Set}$ (i.e. a species) along the inclusion functor $\iota : \mathbb{B} \to \mathbf{Set}$; that is, if there is an isomorphism $$\mathbf{Set}^{\mathbf{Set}}(F,H) \cong \mathbf{Set}^{\mathbb{B}}(G, H\iota)$$ natural in $H$, or, equivalently, if it can be expressed as a coend $$F(A) \cong \int^{C\in\mathbb{B}} \mathbf{Set}(\iota(C), A) \cdot G(C).$$ More intuitively, looking at the coend formula above, one can think of analytic functors as those for which $F(A)$ can be "decomposed" into a labelled shape, i.e. a species structure $G$ with labels taken from some finite set $C$, together with a function mapping labels in $C$ to values in $A$. The isomorphism of hom-sets above then says that natural maps out of $F$ are in correspondence with species morphisms out of $F$'s "underlying shape" $G$.

What are some examples of functors $F : \mathbf{Set} \to \mathbf{Set}$ which are not analytic?

Answer 1

In his article Foncteurs analytiques et espèces de structures (Lecture Notes in Mathematics 1234), Joyal characterizes analytic functors as those which preserve filtered colimits, cofiltered limits, and weak pullbacks. So it's just a matter of finding functors which violate one of these properties.

A functor that preserves filtered colimits is said to be finitary, and it is easy to come up with examples of non-finitary endofunctors on $\mathbf{Set}$. One example is the covariant power set functor $P$; this does not preserve for example the colimit of the filtered diagram $D$ of finite subsets of $\mathbb{N}$ and inclusions between them, by simple cardinality considerations. Another is the endofunctor $F = (-)^\mathbb{N}$, where again the comparison map $\text{colim} \; F D \to F(\mathbb{N})$ is not an isomorphism (the identity map $1_\mathbb{N}$ as an element of $F(\mathbb{N})$ is not in the image of the comparison map).