Calculating Colimits Compositionally

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages.

💡 Research Summary

The paper presents a novel, fully compositional method for calculating finite limits and colimits by exploiting the algebraic structures of spans and cospans. After motivating the need for a modular approach—traditional limit/colimit calculations often require a global view of a diagram—the authors introduce the categorical notions of a span (an object X equipped with morphisms X → A and X → B) and a cospan (an object Y equipped with morphisms A → Y and B → Y). They show that spans form a symmetric monoidal category under pullback composition, while cospans form a symmetric monoidal category under pushout composition. Both categories can be organized into a PROP (product‑and‑permutation category), providing a uniform algebraic framework.

The central theoretical contribution is the proof that the span algebra generates all finite limits and the cospan algebra generates all finite colimits. To achieve this, the authors define two functors on the PROP: a “lifter” that turns spans into limit objects via pullbacks, and a “co‑lifter” that turns cospans into colimit objects via pushouts. These functors satisfy the necessary coherence conditions, allowing any complex diagram to be decomposed into a sequence of elementary span or cospan compositions. Consequently, a global limit or colimit can be reconstructed by iteratively applying pullbacks or pushouts, respectively, without ever leaving the local algebraic setting.

The paper illustrates the technique with concrete examples. For instance, the colimit of a chain of morphisms f: A → B and g: B → C is obtained by first representing each morphism as a cospan, then taking the pushout of the intermediate object B. The same pattern works for binary products, equalizers, and other standard limits, demonstrating that the method scales uniformly across different shapes of diagrams.

A particularly striking application is a categorical proof of Kleene’s theorem on regular languages. By interpreting regular expressions as cospans in the category of sets and relations, the authors construct finite automata via pushout operations that correspond to concatenation, union, and Kleene star. The pushout construction automatically yields the closure properties required by the theorem, providing a clean, algebraic derivation that avoids the usual state‑by‑state construction. This not only re‑validates Kleene’s theorem from a new perspective but also showcases the expressive power of the span/cospan calculus for language‑theoretic problems.

The related‑work discussion positions the contribution against earlier compositional algebra approaches that focused mainly on monoidal or operadic structures. By handling both limits and colimits within a single unified algebra of spans and cospans, the paper achieves a level of symmetry and generality not present in prior work. The authors also outline potential extensions, such as handling infinite limits/colimits, higher‑dimensional diagrams, and applications to type systems, data‑flow analysis, and categorical semantics of programming languages.

In conclusion, the paper delivers a robust, compositional toolkit for categorical constructions, demonstrates its practical utility through a novel proof of Kleene’s theorem, and opens avenues for further research in both theoretical and applied categorical computer science.