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Constructing span categories from categories without pullbacks

David Weisbart and Adam M. Yassine

Vol. 321 (2022), No. 2, 443–465
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 𝒞 and 𝒞 and a functor from 𝒞 to 𝒞, we introduce the notion of an -pullback of a cospan in 𝒞, as well as the notion of span tightness of . If is span tight, then we can form a generalized span category Span (𝒞,) and circumvent the technical difficulty of 𝒞 failing to have pullbacks. Composition in Span (𝒞,) uses -pullbacks rather than pullbacks and in this way differs from the category Span (𝒞), but reduces to it when both 𝒞 has pullbacks and is the identity functor.

In memory of Professor V. S. Varadarajan

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Keywords
span categories, pullbacks, compositionality, fibered products, forgetful functors
Mathematical Subject Classification
Primary: 18F99, 53Z05
Secondary: 70A05
Milestones
Received: 14 September 2020
Accepted: 7 November 2022
Published: 21 March 2023
Authors
David Weisbart
Department of Mathematics
University of California, Riverside
Riverside, CA
United States
Adam M. Yassine
Department of Mathematics
Bowdoin College
Brunswick, ME
United States