论文标题
左伴随函数的逆极限
Inverse limits of left adjoint functors on pointed sets
论文作者
论文摘要
本文是[bash]的延续，在该论文中，我们研究了在逆限制下Abelianization函子的行为。 [bash]我们的主要结果是，如果$ \ mathcal {t} $是可计数的定向poset，$ g：\ nathcal {t} \ to \ nathcal {g} rp $是满足Mittag-Leffler条件的组的图，则是自然图$ \ mathrm {ab}（{\ lim} _ {t \ in \ mathcal {t}} g_t）\ to {\ lim} _ {t \ in \ int \ mathcal {t}}} \ mathrm {ab}（ab}（g_t）$ is usundivie and a kornel is a kortiors and kortion。 Abelianization是从组到Abelian组的左伴函子的一个例子。 在本文中，我们研究了左伴函数的反比限制的行为，从尖头组合到阿贝尔组。此类函数由Abelian组分类，在哪里到Abelian goups $ a $对应于左伴函数$ l_a：\ Mathcal {s} \ text {et} _*\ to \ nathcal {a} a} \ text {a} \ text {B text {b} $由$ l_a（y）= \ bigoplus _ $ \ y $ \ $ \ MATHCAL {T} $是一个定向的poset，$ x：\ Mathcal {t} \ to \ Mathcal {s} \ text {et} _*$是尖的集合的一个图，我们显示自然地图$ pum $ p $ρ：l_a（l_a} {\ lim} _ {\ lim} _ { \ to {\ lim} _ {t \ in \ mathcal {t}} l_a（x_t）$$是iNjective。此外，如果$ \ Mathcal {t} $是可计数的，并且$ x $满足Mittag-Leffler条件，我们表明$ρ$的cokernel是一个代数紧凑的组。与[bash]的主要结果相比，代数紧凑要比cotorsion强得多，因为它也要求ULM长度为$ \ leq 1 $。 我们还表明，即使以弱的结转形式，这种结果也不会扩展到无数的图表。即，如果$ a $不是可划分组和有限组的产物，我们将构建一个定向的poset $ \ nathcal {t} $，带有$ | \ m rathcal {t} | = 2^{\ aleph_0} $，图$ x：\ mathcal { Mittag-Leffler条件，使得$ρ$的Cokernel不是Cotorsion。
This paper is a continuation of [BaSh], where we studied the behaviour of the abelianization functor under inverse limits. Our main result in [BaSh] was that if $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}\to\mathcal{G} rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}({\lim}_{t\in\mathcal{T}}G_t)\to {\lim}_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ is surjective, and its kernel is a cotorsion group. The abelianization is an example of a left adjoint functor from groups to abelian groups. In this paper we study the behaviour under inverse limits of left adjoint functors from pointed sets to abelian groups. Such functors are classified by abelian groups, where to the abelian group $A$ corresponds the left adjoint functor $L_A:\mathcal{S} \text{et}_*\to\mathcal{A} \text{b}$ given by $L_A(Y)=\bigoplus_{Y\setminus\{*\}}A.$ If $\mathcal{T}$ is a directed poset and $X:\mathcal{T}\to\mathcal{S} \text{et}_*$ is a is diagram of pointed sets, we show that the natural map $$ρ: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 $ρ$ 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 $ρ$ is not cotorsion.
