代数精确性的一个特征
A characterisation of algebraic exactness
摘要：
一个代数正合范畴，它允许所有的极限和上极限，而每一个代数簇都具有这些极限和上极限，并且每一个簇之间的遗忘函子都保持这些极限和上极限，并且它证明这些极限和上极限之间的相互作用与任何簇中的相互作用相同。Adámek、Lawvere和Rosicklane研究了这类范畴：他们将它们描述为具有小极限和筛余极限的范畴，其中取筛余极限的函子是连续的。他们证明了一个完备范畴和筛余完备范畴只有当它是Barr-正合范畴、有限极限与滤余极限可换范畴、正则满态对于小乘积是稳定的范畴、滤余极限分布在小乘积上时才是代数正合范畴。我们证明了这个猜想。
英文摘要：
An algebraically exact category in one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Adámek, Lawvere and Rosický: they characterised them as the categories with small limits and sifted colimits for which the functor taking sifted colimits is continuous. They conjectured that a complete and sifted-cocomplete category should be algebraically exact just when it is Barr-exact, finite limits commute with filtered colimits, regular epimorphisms are stable by small products, and filtered colimits distribute over small products. We prove this conjecture.
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