On WordPress you need to add “latex” after the first dollar sign (apparently other blogs need to write ordinary dollar signs sometimes).

]]>Well, I’m certainly happy to believe that the algebras for $x \mapsto 2^{2^x}$ has no initial object. (My math doesn’t seem to process in comments — what am I don’t wrong?) For functors built out of products and coproducts, I think I see how to prove that the category of algebras is presentable, but for these “exponential” functors I will make no such claim.

]]>Interesting example! My intuitions about this might not have been as well-calibrated as I thought.

Groupprops has a cute and reasonably short proof: the idea is just to show that one can cook up a suitable homomorphism into a symmetric group. Geometrically one is cooking up a suitable covering space of a wedge of circles.

]]>I agree with Theo that the residual finiteness of free groups is mysterious. The only proof that I can think of at the moment is to use the ping-pong lemma to show that contains a free subgroup, and then show that itself is residually finite, since any element is non-trivial modulo an appropriate prime $p$. Is there a simpler way to do it?

Moreover, the state of affairs with discrete groups often *is* strange: for example, in finite groups, the equation implies that commutes with , but this implication fails in some infinite groups! This is a classical result of Baumslag and Solitar; see this shameless self-promotion (p.6, proof of Thm 1(a)) for a proof along the lines of Theo’s comment.

Take the double powerset functor . This doesn’t have an initial algebra by Lambek’s theorem and Cantor’s theorem.

Hmm. Maybe.

]]>On an unrelated note, I find the residual finiteness of free groups utterly mysterious. The eigenvalues of any element of any finite group acting on any vector space are roots of unity, and therefore algebraic integers with unit norm (at all places). I could certainly imagine a situation in which some word held for algebraic integers of unit norm, but not for more general numbers.

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