On functors agreeing with the powerset functor on objects and not being isomorphic to it

category-theoryfunctorsyoneda-lemma

Recall the powerset functor $\mathcal{P}:\mathbf{Set} \to \mathbf{Set}$ defined as

  • $\mathcal{P}(X) = 2^{X} = \{U\subseteq X\}$ on objects
  • $\mathcal{P}(f: X \to Y): \mathcal{P}(X) \to \mathcal{P}(Y),\ \ \ \mathcal{P}(f)(U) = f(U)$ on morphisms

Are there functors $\mathcal{F}:\mathbf{Set}\to \mathbf{Set}$ agreeing with $\mathcal{P}$ on objects, i.e. $\mathcal{F}(X) = \mathcal{P}(X)$ for all sets $X$?

This question was answered in "Is power set functor determined by its image on objects?".

However, all functors given there are naturally isomorphic to $\mathcal{P}$.

Question: Are there any such functors which are not naturally isomorphic to $\mathcal{P}$?

Summary of Results

All functors given in the question linked above can be succinctly characterized as follows:

For each set $X$ choose a bijection $r_X\colon \mathcal P(X)\to\mathcal P(X)$. Now let your functor $\mathcal F$ be defined on morphisms $f\colon X\to Y$ by
$$\mathcal F(f) = r_Y\circ \mathcal P(f) \circ r_X^{-1}$$

One non-trivial choice of $r_X$ would be taking complements, i.e. $r_X(U)=X\setminus U$, then $\mathcal Ff(U) = Y\setminus f(X\setminus U)$.

(source: this post by Christoph)

As per my comment on Christoph's post, we can exhibit the natural isomorphism to $\mathcal{P}$ as follows:

$$
\require{AMScd}
\begin{CD}
\mathcal{P}(X) @>{r_X}>> \mathcal{P}(X)\\
@V{\mathcal{P}f}VV @VV{\mathcal{F}f = r_y\ \circ\ \mathcal{P}f\ \circ\ r_X^{-1}}V \\
\mathcal{P}(Y) @>{r_Y}>> \mathcal{P}(Y)\end{CD}
$$

Best Answer

There seems to be quite a lot of literature on functors from sets to sets that are determined, up to isomorphism, by their values on objects ("DVO functors").

I found a paper (reference below) which essentially contained the following example to show that the power set functor is not.

Define a functor $F$ such that $FX=\mathcal{P}(X)$ for every set $X$, and for a function $f:X\to Y$, $Ff:\mathcal{P}(X)\to\mathcal{P}(Y)$ is the function such that, for $A\subseteq X$, $Ff(A)=f(A)$ if the restriction of $f$ to $A$ is injective, and $Ff(A)=\emptyset$ if the restriction of $f$ to $A$ is not injective.

Cancila, Daniela; Honsell, Furio; Lenisa, Marina, Functors determined by values on objects, Brookes, Steve (ed.) et al., Proceedings of the 22nd annual conference on mathematical foundations of programming semantics (MFPS XXII), Genova, Italy, May 23–27, 2006. Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 158, 151-169 (2006).

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