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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 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.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 11th 2014
   • PermaLink
   Author: Chris SchommerPries
   Format: MarkdownItex[globular operads](http://ncatlab.org/nlab/show/globular+operad) are a great way to formalize all kinds of higher categorical structure you can put on globular sets. I am wondering if there is a globular operad corresponding to the theory of symmetric monoidal categories (and similar higher dimensional versions). I am worried because the n-lab page says that the resulting monads on globular sets can be characterized as those which are Cartesian and also over T, the free strict n-category monad. I fear that the symmetric monoidal category monad might fail to be Cartesian for the same reason that the commutative monoid monad on sets fails to be Cartesian. What is to be done about this?

   globular operads are a great way to formalize all kinds of higher categorical structure you can put on globular sets. I am wondering if there is a globular operad corresponding to the theory of symmetric monoidal categories (and similar higher dimensional versions).

   I am worried because the n-lab page says that the resulting monads on globular sets can be characterized as those which are Cartesian and also over T, the free strict n-category monad. I fear that the symmetric monoidal category monad might fail to be Cartesian for the same reason that the commutative monoid monad on sets fails to be Cartesian.

   What is to be done about this?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 11th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think the symmetric monoidal category monad *is* cartesian; it precisely remedies the problem with the commutative monoid monad by causing $a\otimes b$ to be only isomorphic to $b\otimes a$ rather than equal to it.

   I think the symmetric monoidal category monad is cartesian; it precisely remedies the problem with the commutative monoid monad by causing $a\otimes b$ to be only isomorphic to $b\otimes a$ rather than equal to it.

3. • CommentRowNumber3.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 12th 2014
   • PermaLink
   Author: Chris SchommerPries
   Format: MarkdownItexI see Lienster's book mentions something along these lines for symmetric strict monoidal categories. I wish there was an easy way to check this.

   I see Lienster’s book mentions something along these lines for symmetric strict monoidal categories. I wish there was an easy way to check this.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexEasier than just checking the definition?

   Easier than just checking the definition?

5. • CommentRowNumber5.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 12th 2014
   • PermaLink
   Author: Chris SchommerPries
   Format: MarkdownItexWait, I don't think there is a map of monads $P \to T$, where T is the strict 2-category monad and P is the symmetric monoidal bicategory monad. Such a map would mean that every strict 2-category could be viewed as a special case of a symmetric monoidal bicategory, which is absurd. So I guess I am asking if there is some alternative theory to globular operads which is similar but includes this example?

   Wait, I don’t think there is a map of monads $P \to T$, where T is the strict 2-category monad and P is the symmetric monoidal bicategory monad. Such a map would mean that every strict 2-category could be viewed as a special case of a symmetric monoidal bicategory, which is absurd.

   So I guess I am asking if there is some alternative theory to globular operads which is similar but includes this example?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI don't know.

   I don’t know.