论文标题
从没有回调的类别中构建跨度类别
Constructing Span Categories From Categories Without Pullbacks
论文作者
论文摘要
SPAN类别为某些系统的数学模型形式化提供了一个抽象框架。某些系统的数学描述，例如经典的机械系统，需要没有回调的类别，这将跨度类别的效用限制为正式框架。给定类别$ \ MATHSCR {C} $和$ \ MATHSCR {C}^\ Prime $和Founcor $ \ Mathcal F $来自$ \ Mathscr {C} $到$ \ Mathscr {c}^\ prime的$ \ Mathscr {c} $ $ \ Mathcal f $的跨度紧密度。如果$ \ MATHCAL F $跨度紧密，那么我们可以形成一般的跨度类别$ {\ rm span}（\ Mathscr {c}，\ Mathcal f）$，并避免$ \ MATHSCR {C} $的技术难度。 $ {\ rm span}（\ Mathscr {c}，\ Mathcal f）$中的成分使用$ \ Mathcal f $ -pullbacks而不是回溯，而这样的方式与类别$ {\ rm span}（\ rm span}（\ rm span}（\ mathscr {c}）$ n $ nath $ n $ n $ cr cr. f $是身份函数。
Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks, and this limits the utility of span categories as a formal framework. Given categories $\mathscr{C}$ and $\mathscr{C}^\prime$ and a functor $\mathcal F$ from $\mathscr{C}$ to $\mathscr{C}^\prime$, we introduce the notion of an $\mathcal F$ pullback of a cospan in $\mathscr{C}$, as well as the notion of span tightness of $\mathcal F$. If $\mathcal F$ is span tight, then we can form a generalized span category ${\rm Span}(\mathscr{C},\mathcal F)$ and circumvent the technical difficulty of $\mathscr{C}$ failing to have pullbacks. Composition in ${\rm Span}(\mathscr{C},\mathcal F)$ uses $\mathcal F$-pullbacks rather than pullbacks and in this way differs from the category ${\rm Span}(\mathscr{C})$, but reduces to it when both $\mathscr{C}$ has pullbacks and $\mathcal F$ is the identity functor.
