Inverse limits of left adjoint functors on pointed sets

TL;DR

This paper investigates how left adjoint functors from pointed sets to abelian groups behave under inverse limits, establishing conditions for injectivity and describing the nature of cokernels, with implications for algebraic compactness.

Contribution

It extends previous work on abelianization by analyzing inverse limits of left adjoint functors from pointed sets, showing injectivity and characterizing cokernels as algebraically compact groups.

Findings

The natural map is injective for inverse limits of pointed sets.

Cokernels are algebraically compact groups under certain conditions.

The results do not extend to uncountable diagrams without additional assumptions.

Abstract

This paper is a continuation of [BaSh], where we studied the behaviour of the abelianization functor under inverse limits. Our main result in [BaSh] was that if $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}\to\mathcal{G} rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}({\lim}_{t\in\mathcal{T}}G_t)\to {\lim}_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ is surjective, and its kernel is a cotorsion group. The abelianization is an example of a left adjoint functor from groups to abelian groups. In this paper we study the behaviour under inverse limits of left adjoint functors from pointed sets to abelian groups. Such functors are classified by abelian groups, where to the abelian group $A$ corresponds the left adjoint functor $L_A:\mathcal{S} \text{et}_*\to\mathcal{A} \text{b}$ given by $L_A(Y)=\bigoplus_{Y\setminus\{*\}}A.$ If $\mathcal{T}$ is a directed poset and $X:\mathcal{T}\to\mathcal{S} \text{et}_*$ is a is diagram of pointed sets, we show that the natural map $$\rho:L_A({\lim}_{t\in\mathcal{T}}X_t) \to{\lim}_{t\in\mathcal{T}}L_A(X_t)$$ is injective. If, in addition, $\mathcal{T}$ is countable and $X$ satisfies the Mittag-Leffler condition, we show that the cokernel of $\rho$ is an algebraically compact group. Compared with the main result in [BaSh], algebraically compact is much stronger then cotorsion as it also requires the Ulm length to be $\leq 1$. We also show that this result, even in its weak form of cotorsion, does not extend to uncountable diagrams. Namely, if $A$ is not the product of a divisible group and a bounded group, we construct a directed poset $\mathcal{T}$ with $|\mathcal{T}|=2^{\aleph_0}$ and a diagram $X:\mathcal{T}\to\mathcal{S} \text{et}_*$, that satisfies the Mittag-Leffler condition, such that the cokernel of $\rho$ is not cotorsion.