Constructing Span Categories From Categories Without Pullbacks

TL;DR

This paper introduces a generalized span category construction that overcomes the lack of pullbacks in certain categories by using a functor-based approach, expanding the applicability of span categories in mathematical modeling.

Contribution

It defines the notion of an $ ext{F}$-pullback and span tightness, enabling the construction of span categories without requiring pullbacks in the original category.

Findings

Generalized span categories can be formed using $ ext{F}$-pullbacks.

The construction reduces to classical span categories when pullbacks exist.

This approach broadens the use of span categories in systems lacking pullbacks.

Abstract

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.