论文标题
单个模块类别
Univalent categories of modules
论文作者
论文摘要
我们显示，同型类型理论（HOTT）中环上的模块类别满足同源代数的AB公理的内部版本。主要的微妙之处在于证明AB4，即由任意集索引的索引是左撇子。为了证明这一点，我们用严格的（有序）有限子媒体组的类别替换了一个集合。通过表明后者被过滤，我们推断了相关的左脱位。更普遍地，我们表明过滤colimits（AB5）的精确性意味着Hott中任何Abelian类别的AB4。 Roswitha Harting在具有自然数对象的小学拓扑群中对Abelian群体的内部共同构造的构建受到了极大的启发[HAR82]。 为了陈述AB公理，我们定义和研究在Hott中过滤（和筛选）的预定剂。需要的关键结果是滤波的colimits上通勤，并具有有限的集合限制。这是一个熟悉的经典结果，但以前尚未在我们的环境中检查过。 最后，我们将最中心的结果解释为$ \ infty $ -TOPOS $ \ MATHSCR {X} $。给定一个环$ r $ in $ \ mathscr {x} $，我们表明$ r $ -modules的内部类别in $ \ mathscr {x} $代表将对象发送$ x \ in \ mathscr {x} $的$ x {x {x {x {\ times} r）$ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x。通常，我们的结果产生了产品保护的左伴随，以使模块的基本变化超过$ x $。当$ x $是$ 0 $ - 截断时，此左伴随是内部互换。通过内部化过程，我们从HOTT的内部左置表中推断出内部副函数的左脱位。
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 with its strict category of (ordered) finite sub-multisets. 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 [Har82]. 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 $\mathscr{X}$, 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 over $X$. 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.
