Monoid-like definitions of cyclic operads

TL;DR

This paper introduces two monoid-like definitions of cyclic operads within Joyal's species framework, demonstrating their equivalence and connecting different characterizations of cyclic operads.

Contribution

It provides novel monoid-like formulations for cyclic operads and proves their equivalence, bridging two existing characterizations within a categorical setting.

Findings

Two monoid-like definitions of cyclic operads are introduced.

The equivalence between the 'exchangable-output' and 'entries-only' characterizations is established.

The approach relies on descent for species and categorical methods.

Abstract

Guided by the microcosm principle of Baez-Dolan and by the algebraic definitions of operads of Kelly and Fiore, we introduce two "monoid-like" definitions of cyclic operads, one for the original, "exchangable-output" characterisation of Getzler-Kapranov, and the other for the alternative "entries-only" characterisation, both within the category of Joyal's species of structures. Relying on a result of Lamarche on descent for species, we use these "monoid-like" definitions to prove the equivalence between the "exchangable-output" and "entries-only" points of view on cyclic operads.