# 2-Segal objects and algebras in spans

## TL;DR

This paper introduces a new categorical framework connecting 2-Segal objects with algebraic structures in spans, establishing equivalences that relate simplicial and cyclic objects to Calabi-Yau algebras.

## Contribution

It defines a category for Calabi-Yau algebra objects in spans and proves key equivalences linking 2-Segal objects to algebraic structures in this setting.

## Key findings

- Establishment of an equivalence between 2-Segal simplicial objects and algebra objects in spans.
- Demonstration that 2-Segal cyclic objects correspond to Calabi-Yau algebra objects in spans.
- Development of a categorical framework unifying these algebraic and combinatorial structures.

## Abstract

We define a category parameterizing Calabi-Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi-Yau algebra objects in Span(C).

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Source: https://tomesphere.com/paper/1905.06671