作为寓言的跨度范畴的商和规则范畴的表示
Quotients of span categories that are allegories and the representation of regular categories
摘要：
我们考虑范畴C中的态射的跨距（的同构类）的普通范畴Span（C），它具有有限的极限，通过拉回水平组合，并给出Span（C）的商是寓言的一般判据.特别地，当C带有拉回稳定的但不一定是真的（E，M）-分解系统时，我们建立了一个与C中M-关系范畴Rel_M（C）同构的商范畴Span_E（C），并证明了当M是C中的单态映射类时，它是一个（酉的和表式的）寓言.如果没有这个限制，仍然可以找到一个最小拉回稳定和组合封闭的类E。其中，E是E，(C)是一个单一的表格式寓言。用这种方法，我们得到了2-函子的一个左伴随，它给每一个酉表寓言分配了它的Lawverian映射的正则范畴。利用正则范畴的Fredom-Scedrov表示定理，我们得到了每一个具有稳定分解系统的完全范畴都有一个到正则范畴的巨2-范畴的反射.
英文摘要：
We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E, M)-factorization system, we establish a quotient category Span_E(C) that is isomorphic to the category Rel_M(C) of M-relations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can still find a least pullback-stable and composition-closed class E. containing E such that Span_E.(C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2-category of all regular categories.
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