算术范畴的新刻画与Pixley定理
A new characterization of arithmetical categories and Pixley's theorem
AI总结：
本文通过推出与极限的比较态射条件刻画算术范畴，证明其等价于精确Mal'tsev性，并基于中国剩余定理给出Pixley定理的新证明。
AI中文摘要：
我们用某些适当的推出和一个有限极限上的简单条件来刻画算术范畴——即同余格是分配格的精确Mal'tsev范畴：一个正则范畴是精确算术范畴当且仅当，对于每一组具有共同定义域的正则满态射三元组，两两推出存在，且到它们所形成图的极限的比较态射是正则满态射。对于正则范畴，该条件对二元组和三元组成立等价于对所有$n \ge 2$的$n$元组成立，并且在正则范畴中刻画了算术范畴。仅对二元组而言，该条件归结为Carboni、Kelly和Pedicchio所给出的正则范畴中精确Mal'tsev范畴的刻画。对于$n \ge 4$的$n$元组，不再得到更多：算术性是该阶梯的第三级也是最后一级。证明依赖于中国剩余定理的范畴形式；在精确Mal'tsev语境中，该定理的这种形式接近于Hoefnagel关于多数范畴的结果。作为直接应用，我们利用适当的自由代数图给出了Pixley关于算术簇的刻画的一个新证明。
英文摘要：
We characterize the arithmetical categories - the exact Mal'tsev categories whose lattices of congruences are distributive - by a simple condition on some suitable pushouts and a finite limit: a regular category is exact arithmetical if and only if, for every triple of regular epimorphisms with common domain, the pairwise pushouts exist and the comparison morphism to the limit of the diagram they form is a regular epimorphism. For a regular category, the validity of this condition for pairs and for triples is equivalent to its validity for $n$-tuples for all $n \ge 2$, and characterizes arithmetical categories among regular ones. For pairs alone the condition reduces to the characterization of exact Mal'tsev categories among regular ones, due to Carboni, Kelly and Pedicchio. For $n$-tuples with $n \ge 4$ nothing further is obtained: arithmeticity is the third and last rung of that ladder. The proof rests on a categorical form of the Chinese Remainder Theorem; in the exact Mal'tsev context this form of the theorem lies close to results of Hoefnagel on majority categories. As a direct application, we give a new proof of Pixley's characterization of arithmetical varieties using a suitable diagram of free algebras.
