Simple-minded coherence of tricategories Ask Question. My question concerns known results about such “simple-minded” coherence for monoidal bicategories ie. The category of opetopes and the category of opetopic sets. Non-specialist for non-specialist preprints click here. Distributive laws for Lawvere Theories, Submitted book Higher dimensional categories:
Here I am using the algebraic definition of a monoidal bicategory, ie. In Journal of Pure and Applied Algebra , 2: In particular, any bicategory is a category-enriched graph. With Tom Leinster, The question is answered in a paper of Nick Gurski, “An algebraic theory of tricategories” and probably also in his new book “Coherence in Three-dimensional Category Theory”. I stumbled upon this type of questions while studying possible definitions of a dual pair of objects in a monoidal bicategory. I frequently find it very problematic to prove any uniqueness results due to the relevant computations being difficult.
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This is certainly an important and powerful result. This is related to the fact that a one-object monoidal bicategory is “morally the same” as a braided monoidal category a result due to Gordon, Power, Streetwith the braiding given by a clever composition of 2-cells. Hence, I am looking for techniques that could simplify working with a general tricategory.
Any tricategory is triequivalent to a Gray -category, ie.
Research – Nick Gurski
Unicorn Meta Zoo 3: With Nick Gurski and Emily Riehl. Eugenia Cheng’s Research papers The category of opetopes and the category of opetopic sets.
A note on Penon’s definition of weak n -category. Sign up using Email and Password. In Journal of Pure and Applied Algebra, 3: Weak omega-categories via terminal coalgebras.
The periodic table of n -categories for mick dimensions II: In Theory and Applications of Categories, With Tom Leinster, Theory and Applications of Categories 29 Cyclic multicategories, multivariable adjunctions and mates. In Theory and Applications of Categories, Here I say “simple-minded” to mean that it can be presented as a statement of the form “some diagram commutes”.
With Aaron Lauda, Cyclic multicategories, multivariable adjunctions and mates. In particular the “naive” version of coherence for monoidal bicategories I asked thwsis above is true. The strictifying version of coherence is an important theorem on its own right, but it also implies the simple-minded version of coherence with the following argument.
In The Princeton Companion to Mathematicsed.
Research – Eugenia Cheng
In Applied Categorical Structures15 4: Sign up or log in Sign up using Google. Recall that coherence for tricategories as proved by Gordon, Street, Power has the following form:.
Sign up using Facebook. A direct nivk that the category of 3-computads is not cartesian closed. In particular, any bicategory is a category-enriched graph. With Aaron Lauda, In Homotopy, Homology and Applications13 2: As braiding are in general nock symmetries, some diagrams of constraint 2-cells in monoidal bicategories do not commute in general. ImaginaryBerlin July, Invited speaker. Is there any general framework for proving that some classes of diagrams commute in every tricategory?
With Tom Leinster, Also available hereand on the arXiv