# Span composition using fake pullbacks

## TL;DR

This paper introduces a method for composing spans in categories lacking traditional pullbacks by utilizing fake pullbacks, extending the construction to more general settings than Puppe-exact categories.

## Contribution

It demonstrates how spans of EM-spans can be composed using fake pullbacks, broadening the scope of span composition in categories without actual pullbacks.

## Key findings

- Spans of EM-spans admit fake pullbacks for composition.
- The approach generalizes the construction beyond Puppe-exact categories.
- Provides a new perspective on span composition in non-traditional categories.

## Abstract

The construction of a category of spans can be made in some categories $\CC$ which do not have pullbacks in the traditional sense. The PROP for monoids is a good example of such a $\CC$. The 2012 book concerning homological algebra by Marco Grandis gives the proof of associativity of relations in a Puppe-exact category based on a 1967 paper of M.\v{S}. Calenko. The proof here is a restructuring of that proof in the spirit of the first sentence of this Abstract. We observe that these relations are spans of EM-spans and that EM-spans admit fake pullbacks so that spans of EM-spans compose. Our setting is more general than Puppe-exact categories.