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?



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


