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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.
   • CommentAuthorUrs
   • CommentTimeDec 7th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI extracted the definition of "n-category with all duals" from Scott Morrison and Kevin Walker's "Blob homology" at [[blob n-category]]. This is to some extent a take on defining [[hyperstructure]]s.

   I extracted the definition of “n-category with all duals” from Scott Morrison and Kevin Walker’s “Blob homology” at blob n-category.

   This is to some extent a take on defining hyperstructures.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 7th 2010
   • (edited Dec 7th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded examples: fundamental blob $n$-category of a topological space and bordism $n$-category

   added examples: fundamental blob $n$-category of a topological space and bordism $n$-category

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeDec 8th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIn the definition of $\underset{\to}{C}_{k-1}(\partial X)$, is the colimit over the *groupoid* of balls in the (k-1)-sphere? That doesn't seem right, but it's the only thing I see how to define, since $C_{k-1}$ is only defined on that groupoid. I couldn't find where in their paper they define $\underset{\to}{C}$.

   In the definition of $\underset{\to}{C}_{k-1}(\partial X)$, is the colimit over the groupoid of balls in the (k-1)-sphere? That doesn’t seem right, but it’s the only thing I see how to define, since $C_{k-1}$ is only defined on that groupoid. I couldn’t find where in their paper they define $\underset{\to}{C}$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2010
   • (edited Dec 8th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis is around def. 6.3.2 in their article. I have added a liitle bit more detail to the entry [[blob n-category]], but still not everything.

   This is around def. 6.3.2 in their article. I have added a liitle bit more detail to the entry blob n-category, but still not everything.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeDec 8th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThis definition seems very circular! The "boundary restriction" transformation is defined in terms of the extension to general shapes, which is defined in terms of the composition operation, which uses the boundary restriction maps to define when things are composable. Is there some implicit induction on dimensions going on?

   This definition seems very circular! The “boundary restriction” transformation is defined in terms of the extension to general shapes, which is defined in terms of the composition operation, which uses the boundary restriction maps to define when things are composable. Is there some implicit induction on dimensions going on?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2010
   • (edited Dec 8th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> This definition seems very circular! Yes, indeed. And the fact that the writeup is very non-linear itself makes it hard to plough through. I _think_ that, as you suggest, we are supposed to define this inductively: first we define composition of 1-morphisms/1-blobs, that allows us to extend the blob $n$-graph to 1-spheres, then using this we define compostion of 2-morphisms/2-blobs, and so forth.

   This definition seems very circular!

   Yes, indeed. And the fact that the writeup is very non-linear itself makes it hard to plough through.

   I think that, as you suggest, we are supposed to define this inductively: first we define composition of 1-morphisms/1-blobs, that allows us to extend the blob $n$-graph to 1-spheres, then using this we define compostion of 2-morphisms/2-blobs, and so forth.

7. • CommentRowNumber7.
   • CommentAuthorkwalker
   • CommentTimeDec 9th 2010
   • PermaLink
   Author: kwalker
   Format: MarkdownItexAs Urs said, the category definition is spirally, not circular. The colimit construction which extends from balls to any manifold (in particular, to spheres) is described in section 6.3. To state the "boundary restriction" axiom for k-morphsims requires all of the axioms for (k-1)-morphisms. So if we were presenting things in linear order, we would give all the axioms for 0-morphsims, then all the axioms for 1-morphisms, and so on. The colimit which defines $\underset{\to}{C}_{k-1}(\partial X)$ is over the poset of all decompositions of $\partial X$ into (k-1)-balls.

   As Urs said, the category definition is spirally, not circular. The colimit construction which extends from balls to any manifold (in particular, to spheres) is described in section 6.3. To state the “boundary restriction” axiom for k-morphsims requires all of the axioms for (k-1)-morphisms. So if we were presenting things in linear order, we would give all the axioms for 0-morphsims, then all the axioms for 1-morphisms, and so on.

   The colimit which defines $\underset{\to}{C}_{k-1}(\partial X)$ is over the poset of all decompositions of $\partial X$ into (k-1)-balls.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeDec 9th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have edited the entry [[blob n-category]] a little to reflect this fact better. But Idon't quite have the time right now to add more details.

   I have edited the entry blob n-category a little to reflect this fact better.

   But Idon’t quite have the time right now to add more details.