The notion of an $\mathcal{M}$-coextensive object is introduced in an arbitrary category $\mathbb{C}$, where $\mathcal{M}$ is a distinguished class of morphisms from $\mathbb{C}$. This notion allows for a categorical treatment of the strict refinement property in universal algebra, and highlights its connection with extensivity in the sense of Carboni, Lack and Walters. If $\mathcal{M}$ is the class of all product projections in a variety of algebras $\mathbb{C}$, then the $\mathcal{M}$-coextensive (or projection-coextensive) objects in $\mathbb{C}$ turn out to be precisely those algebras which have the strict refinement property. If $\mathcal{M}$ is the class of surjective homomorphisms in the variety, then the $\mathcal{M}$-coextensive objects are precisely those algebras which have directly-decomposable (or factorable) congruences. In exact Mal'tsev categories, every centerless object with global support has the strict refinement property. We will also show that in every exact majority category, every object with global support has the strict refinement property.

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M-同延对象和严格细化性质

$\mathcal{M}$-coextensive object 的概念被引入到任意范畴 $\mathbb{C}$ 中，其中 $\mathcal{M}$ 是来自 $\mathbb{C}$ 的一个特异类态射. 这个概念允许对泛代数中的严格细化性质进行分类处理，并强调其与 Carboni、Lack 和 Walters 意义上的可扩展性的联系。如果 $\mathcal{M}$ 是各种代数 $\mathbb{C}$ 中所有乘积投影的类，则 $\mathcal{M}$-coextensive（或projection-coextensive）对象在 $\mathbb {C}$ 恰好是那些具有严格细化性质的代数。如果 $\mathcal{M}$ 是簇中的满射同态类，那么 $\mathcal{M}$-coextensive 对象正是那些具有可直接分解（或可分解）同余的代数。在精确的 Mal'tsev 类别中，每个具有全局支持的无心对象都具有严格的细化性质。我们还将证明，在每个精确多数类别中，每个具有全局支持的对象都具有严格的细化属性。