Models for cyclic infinity operads

In their paper "Models for cyclic infinity operads," authors Brandon Doherty and Philip Hackney introduce model structures on cyclic dendroidal sets and cyclic dendroidal spaces for cyclic quasi-operads and complete cyclic dendroidal Segal spaces. They demonstrate that these models are Quillen equivalent to the model structure for simplicial cyclic operads, providing a significant advancement in the field of algebraic topology. The authors confirm the existence of such structures by addressing a question posed by the second author and Drummond-Cole regarding model structures for cyclic $\infty$-operads. Furthermore, they lay the groundwork for further research in this area by extending their analysis to planar cyclic $\infty$-operads. This work not only contributes to the theoretical framework of cyclic 2-Segal spaces but also offers practical implications for understanding the relationship between different types of operads. With 41 pages and 7 figures, this paper provides a comprehensive exploration of model-categorical foundations essential for advancing our knowledge of higher-dimensional algebraic structures.

In their paper "Models for cyclic infinity operads," Doherty and Hackney delve into the world of algebraic topology, exploring the concept of cyclic infinity operads and providing a significant advancement in this field. The authors introduce model structures on cyclic dendroidal sets and spaces, demonstrating their Quillen equivalence to the model structure for simplicial cyclic operads. This not only confirms the existence of such structures but also lays the groundwork for further research in this area.

The paper begins by addressing a question posed by Hackney and Drummond-Cole regarding model structures for cyclic $\infty$-operads. This question stems from previous work done by Lurie, who introduced an $\infty$-categorical framework for studying higher-dimensional algebraic structures known as $\infty$-categories. However, his framework did not include a notion of cyclicity, leading to questions about how to extend it to include this important aspect.

Doherty and Hackney's paper provides an answer to this question by introducing two new model structures: one on cyclic dendroidal sets and another on cyclic dendroidal spaces. These models are designed specifically for studying cyclic quasi-operads (a generalization of ordinary operads) and complete cyclic dendroidal Segal spaces (a type of object used in higher category theory). By showing that these models are Quillen equivalent to the existing model structure for simplicial cyclic operads, they provide evidence that these new models are indeed suitable tools for studying cyclicity.

One key contribution of this paper is its exploration of planar cyclic $\infty$-operads. Building upon their initial work with quasi-operads and Segal spaces, Doherty and Hackney extend their analysis to include planar versions of these objects. Planar objects have been studied extensively in other areas of mathematics such as knot theory, so incorporating them into the study of cyclic $\infty$-operads opens up new avenues for research and potential connections to other fields.

The authors also provide a detailed overview of the model-categorical foundations necessary for understanding their work. This includes discussions on homotopy theory, simplicial sets, and operads. They also introduce the concept of dendroidal sets, which are used as a framework for studying higher-dimensional algebraic structures in this paper.

To illustrate their ideas, Doherty and Hackney include seven figures throughout the paper. These figures help to visualize key concepts and make the material more accessible to readers. Additionally, they provide numerous examples throughout the paper to demonstrate how their models can be applied in practice.

Overall, this paper provides a comprehensive exploration of model-categorical foundations essential for advancing our knowledge of higher-dimensional algebraic structures. By introducing new model structures on cyclic dendroidal sets and spaces and demonstrating their Quillen equivalence to existing models, Doherty and Hackney have made significant contributions to the field of algebraic topology. Their work not only expands our theoretical understanding but also has practical implications for understanding the relationship between different types of operads. With 41 pages packed with valuable insights and analysis, "Models for cyclic infinity operads" is an important read for anyone interested in higher-dimensional algebraic structures or algebraic topology in general.