arXiv subjects
Publications and source records attributed to Jovana Obradovic.
We introduce a diagrammatic language for compact, orientable 3-dimensional manifolds with boundary. A diagrammatic calculus (both integral and rational version) appropriate for this language is introduced and its completeness is proved in the paper. Moreover, a calculus consisting of a finite list of local moves is presented.
A formal sequent system dealing with Menelaus' configurations is introduced in this paper. The axiomatic sequents of the system stem from 2-cycles of Delta-complexes. The Euclidean and projective interpretations of the sequents are defined and a soundness result is proved. This system is decidable and its provable sequents deliver incidence results. A cyclic operad structure tied to this system is presented by generators and relations.
In this paper, we introduce a notion of categorified cyclic operad for set-based cyclic operads with symmetries. Our categorification is obtained by relaxing defining axioms of cyclic operads to isomorphisms and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form "all diagrams of canonical isomorphisms commute". Our coherence results come in two flavours, corresponding to the "entries-only" and "exchangeable-output" definitions of cyclic operads. Our proof of coherence in the entries-only style is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Došen and Petrić. We obtain the coherence in the exchangeable-output style by "lifting" the equivalence between entries-only and exchangeable-output cyclic operads, set up by the second author. Finally, we show that a generalisation of the structure of profunctors of B\' enabou provides an example of categorified cyclic operad, and we exploit the coherence of categorified cyclic operads in proving that the Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd version.
This paper introduces an inductively defined tree notation for all the faces of polytopes arising from a simplex by truncations. This notation allows us to view inclusion of faces as the process of contracting tree edges. Our notation instantiates to the well-known notations for the faces of associahedra and permutohedra. Various authors have independently introduced combinatorial tools for describing such polytopes. We build on the particular approach developed by Dosen and Petric, who used the formalism of hypergraphs to describe the interval of polytopes from the simplex to the permutohedron. This interval was further stretched by Petric to allow truncations of faces that are themselves obtained by truncations, and iteratively so. Our notation applies to all these polytopes. We illustrate this by showing that it instantiates to a notation for the faces of the permutohedron-based associahedra, that consists of parenthesised words with holes. Dosen and Petric have exhibited some families of hypergraph polytopes (associahedra, permutohedra, and hemiassociahedra) describing the coherences, and the coherences between coherences etc., arising by weakening sequential and parallel associativity of operadic composition. We complement their work with a criterion allowing us to recover the information whether edges of these "operadic polytopes" come from sequential, or from parallel associativity. We also give alternative proofs for some of the original results of Dosen and Petric.