Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 1st 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI've been inactive here for some months now; I hope this will significantly change soon. I have written a stubby beginning of [[iterated monoidal category]], with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions: * Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact? * Assuming the definition is correct, it hinges on the notion of _normal_ lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality? In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products $C$, and form a new 2-category with 2-products $Mon(C)$ whose 0-cells are pseudomonoids in $C$, whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate $Mon(-)$, starting with $C = Cat$. Why isn't this the "right" notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace $Mon(-)$ with $Mon_{norm}(-)$ (where all the units are forced to coincide up to isomorphism)? Apologies if these are naive questions; I am not very familiar with the literature.

   I’ve been inactive here for some months now; I hope this will significantly change soon.

   I have written a stubby beginning of iterated monoidal category, with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions:

   • Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact?

   • Assuming the definition is correct, it hinges on the notion of normal lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality?

   In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products $C$, and form a new 2-category with 2-products $Mon(C)$ whose 0-cells are pseudomonoids in $C$, whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate $Mon(-)$, starting with $C = Cat$. Why isn’t this the “right” notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace $Mon(-)$ with $Mon_{norm}(-)$ (where all the units are forced to coincide up to isomorphism)?

   Apologies if these are naive questions; I am not very familiar with the literature.