Undirected categorical structures

The arrows in a category are obviously directed. What are the undirected analogues of categories and other categorical structures? This page surveys some possible answers.

Dagger categories

A dagger category is a category \(\mathcal{C}\) equipped with an identity-on-objects involutive functor \(\dagger: \mathcal{C}^{\mathrm{op}} \to \mathcal{C}\). Just as every category has an underlying directed graph, every dagger category has an underlying symmetric graph. Symmetric graphs are nearly the same as undirected graphs.

  • Selinger, 2007: Dagger compact closed categories and completely positive maps (doi, pdf)
  • Selinger, 2010: A survey of graphical languages for monoidal categories, Section 7: Dagger categories (doi, arxiv)
  • Karvonen, 2019, PhD thesis: The way of the dagger (arxiv)
    • Ch. 4 develops a theory of limits for dagger categories, based on: Heunen & Karvonen, 2019: Limits in dagger categories (pdf, arxiv)
    • Ch. 5 studies biproducts in categories from the viewpoint of dagger categories, based on: Karvonen, 2020: Biproducts without pointedness (arxiv)
    • Ch. 6 studies dagger Frobenius monads, based on: Heunen & Karvonen, 2016: Monads on dagger categories (pdf, arxiv)

Both of the standard categorical axiomatizations of relations, allegories and bicategories of relations, are dagger categories. The dagger structure is primitive in the former and derived in the latter.

Cyclic operads

A cyclic operad is an operad in which the inputs are indistinguishable from the output. Cyclic operads are to operads as unrooted trees are to rooted trees.

  • Getzler & Kapranov, 1995: Cyclic operads and cyclic homology (pdf)
    • First introduced the term “cylic operad”
    • Unofficial errata at MathOverflow
  • Obradovic, 2017: Monoid-like definitions of cyclic operad (pdf, arxiv)
    • Obradovic, 2017: A formal language for cyclic operads (pdf)
    • Talk at CT 2015 (abstract , slides)
  • Strumila, 2018, talk notes: Asteroidal sets (pdf)
    • Overview of colored cyclic operads with comparison to dagger categories

Modular operads , a special kind of cyclic operad, have been studied by mathematical physicists.