Univalent categories of modules

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-02-01 DOI:10.1017/s0960129523000178
J. G. T. Flaten
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Abstract

We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies in proving AB4, which is that coproducts indexed by arbitrary sets are left-exact. To prove this, we replace a set X with the strict category of lists of elements in X. From showing that the latter is filtered, we deduce left-exactness of the coproduct. More generally, we show that exactness of filtered colimits (AB5) implies AB4 for any abelian category in HoTT. Our approach is heavily inspired by Roswitha Harting’s construction of the internal coproduct of abelian groups in an elementary topos with a natural numbers object. To state the AB axioms, we define and study filtered (and sifted) precategories in HoTT. A key result needed is that filtered colimits commute with finite limits of sets. This is a familiar classical result but has not previously been checked in our setting. Finally, we interpret our most central results into an $\infty$ -topos $ {\mathscr{X}} $ . Given a ring R in $ {\tau_{\leq 0}({{\mathscr{X}}})} $ – for example, an ordinary sheaf of rings – we show that the internal category of R-modules in $ {\mathscr{X}} $ represents the presheaf which sends an object $ X \in {\mathscr{X}} $ to the category of $ (X{\times}R) $ -modules in ${\mathscr{X}} / X$ . In general, our results yield a product-preserving left adjoint to base change of modules over X. When X is 0-truncated, this left adjoint is the internal coproduct. By an internalisation procedure, we deduce left-exactness of the internal coproduct as an ordinary functor from its internal left-exactness coming from HoTT.
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模的单价范畴
证明了同伦类型理论(HoTT)中环上模的范畴满足同伦代数中AB公理的内部版本。主要的微妙之处在于证明AB4,即由任意集合索引的余积是左精确的。为了证明这一点,我们将集合X替换为X中元素列表的严格范畴。通过证明后者是过滤的,我们推导出了余积的左精确性。更一般地,我们证明了对HoTT中任何阿贝尔范畴,滤波边界的精确性(AB5)意味着AB4。我们的方法很大程度上受到Roswitha Harting在具有自然数对象的初等拓扑中构造阿贝群的内副积的启发。为了说明AB公理,我们定义和研究HoTT中的过滤(和筛选)预范畴。需要的一个关键结果是,过滤的边界与有限的集合的极限交换。这是一个熟悉的经典结果,但以前没有在我们的设置中检查过。最后,我们将最核心的结果解释为$\infty$ -topos $ {\mathscr{X}} $。给定$ {\tau_{\leq 0}({{\mathscr{X}}})} $中的一个环R——例如,一个普通的环束——我们表明,$ {\mathscr{X}} $中R-modules的内部类别表示将对象$ X \in {\mathscr{X}} $发送到${\mathscr{X}} / X$中$ (X{\times}R) $ -modules的类别的presheaf。一般来说,我们的结果产生了模在X上的基变化的保积左伴随。当X被截断为0时,这个左伴随是内副积。通过内部化过程,我们从HoTT的内左精确性推导出普通函子的内副积的左精确性。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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