Infinite limit of finitary functors

Question

I'm learning about finitary functors. I have two conflicting results:

1. $F_A X = X^A$ is finitary iff $A$ is finite. Therefore, $X^\mathbb{N}$ is not finitary.
2. $F X = X$ is finitary and the category of finitary functors has small limits (it is a presheaves category). Therefore, $X^\mathbb{N}$ is finitary, because it is the small limit of finitary functors (the product diagram over $\mathbb{N}$ as a discrete category).

What's happening?!

Answer 1

The functor $X \mapsto X^{\mathbb{N}}$ is not the product of $\mathbb{N}$ copies of the functor $X \mapsto X$ in the category of finitary functors $\text{Set} \to \text{Set}$ (it can't be since this functor is not finitary). This product exists but it is something else: namely, thinking of $X^{\mathbb{N}}$ as the set of functions $\mathbb{N} \to X$, it's the set of functions $\mathbb{N} \to X$ which only take finitely many values. (Hence the inclusion of finitary functors into all functors doesn't preserve infinite limits.)

Here are some reasons you might have been confused:

• Given a finitary functor $F : \text{Set} \to \text{Set}$, you can restrict it to a functor $\text{FinSet} \to \text{Set}$, and this produces an equivalence of categories between the functor category $[\text{FinSet}, \text{Set}]$ and the category of finitary functors.
• Also, limits in this functor category exist and are computed pointwise, and so the product of $\mathbb{N}$ copies of the functor $X \mapsto X$, viewed as a functor $\text{FinSet} \to \text{Set}$, is in fact $X \mapsto X^{\mathbb{N}}$.

But! The corresponding finitary functor $\text{Set} \to \text{Set}$ is computed by taking a filtered colimit over all finite subsets of $X$ (which it necessarily must be in order for the result to be finitary), and if you compute that filtered colimit you get the functor above on $\text{Set}$, not $X \mapsto X^{\mathbb{N}}$.

Answer 2

What's happening is that given a small category $\mathcal{C}$, the category of finitary functors is not a presheaf category, but rather a full subcategory of the functor category $[\mathcal{C},\mathbf{Set}]$. It follows rather easily from the fact that filtered colimits commute with finite limits in $\mathbf{Set}$ that this subcategory is closed under finite limits in $[\mathcal{C},\mathbf{Set}]$, but there is no reason for this to extend to small limits.