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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 14th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[(infinity,n)-category of spans]] a pointer to the discussion of $(\infty,2)$-categories of spans in section 10 of * Tobias Dyckerhoff, [[Mikhail Kapranov]], _Higher Segal spaces I_, ([arxiv:1212.3563](http://arxiv.org/abs/1212.3563))

   added to (infinity,n)-category of spans a pointer to the discussion of $(\infty,2)$-categories of spans in section 10 of

   • Tobias Dyckerhoff, Mikhail Kapranov, Higher Segal spaces I, (arxiv:1212.3563)
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 7th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded in the _[Definition](http://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+of+spans#Definition)_ section a brief mentioning of the observation in * [[David Ben-Zvi]], [[David Nadler]], _Nonlinear traces_ ([arXiv:1305.7175](http://arxiv.org/abs/1305.7175)) that $$ Span_n(\mathbf{H}) \simeq E_n Alg_b(\mathbf{H}^{op}) \,. $$

   added in the Definition section a brief mentioning of the observation in

   • David Ben-Zvi, David Nadler, Nonlinear traces (arXiv:1305.7175)

   that

   $$Span_n(\mathbf{H}) \simeq E_n Alg_b(\mathbf{H}^{op}) \,.$$
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 14th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a [brief comment](http://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+of+spans#DefinitionViaCoalgebras) on how to see the above in an elementary way at least for just 1-fold correspondences.

   added a brief comment on how to see the above in an elementary way at least for just 1-fold correspondences.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 7th 2013
   • (edited Aug 8th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI need to think more about the following, but this thought is haunting me, so I thought I'd see if anyone can help me with a comment. For any category ($\infty$-category) $\mathcal{C}$, write $\mathcal{C}^{\Box^n}$ for the category of $n$-dimensional cube diagrams in $\mathcal{C}$. This is naturally an $n$-category ($(\infty,n)$-category) under pasting of cubes. Moreover for $\mathcal{C}$ a category ($\infty$-category) with pullbacks, write $Corr(\mathcal{C})$ for the category ($\infty$-category) of objects and correspondences (spans) in $\mathcal{C}$. Now consider the $n$-category ($(\infty,n)$-category) $$ Corr(\mathcal{C})^{\Box^n} $$ of $n$-fold cubes with edges made of correspondences in $\mathcal{C}$. The thought that is haunting me is: Isn't that much like what we would want anything denoted "$Corr_n(\mathcal{C})$" to be? Heuristically, an $n$-cell in $Corr_n(\mathcal{C})$ is a bunch of objects, with correspondences between them, correspondences of correspondences between those, and so on, up to $n$-fold correspondences. But that vague description certainly also applies to $Corr(\mathcal{C})^{\Box^n}$. You could help me if you can set me straight and point out a good reason why considering $Corr(\mathcal{C})^{\Box^n}$ as "the $n$-category of $n$-fold correspondences in $\mathcal{C}$" is a _bad idea_ . If it indeed is.

   I need to think more about the following, but this thought is haunting me, so I thought I’d see if anyone can help me with a comment.

   For any category ($\infty$-category) $\mathcal{C}$, write $\mathcal{C}^{\Box^n}$ for the category of $n$-dimensional cube diagrams in $\mathcal{C}$. This is naturally an $n$-category ($(\infty,n)$-category) under pasting of cubes.

   Moreover for $\mathcal{C}$ a category ($\infty$-category) with pullbacks, write $Corr(\mathcal{C})$ for the category ($\infty$-category) of objects and correspondences (spans) in $\mathcal{C}$.

   Now consider the $n$-category ($(\infty,n)$-category)

   $$Corr(\mathcal{C})^{\Box^n}$$

   of $n$-fold cubes with edges made of correspondences in $\mathcal{C}$.

   The thought that is haunting me is:

   Isn’t that much like what we would want anything denoted “$Corr_n(\mathcal{C})$” to be?

   Heuristically, an $n$-cell in $Corr_n(\mathcal{C})$ is a bunch of objects, with correspondences between them, correspondences of correspondences between those, and so on, up to $n$-fold correspondences.

   But that vague description certainly also applies to $Corr(\mathcal{C})^{\Box^n}$.

   You could help me if you can set me straight and point out a good reason why considering $Corr(\mathcal{C})^{\Box^n}$ as “the $n$-category of $n$-fold correspondences in $\mathcal{C}$” is a bad idea . If it indeed is.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeAug 8th 2013
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   Author: Mike Shulman
   Format: MarkdownItexCan you explain more about what is "the category of $n$-dimensional cube diagrams in $\mathcal{C}$"?

   Can you explain more about what is “the category of $n$-dimensional cube diagrams in $\mathcal{C}$”?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeAug 8th 2013
   • (edited Aug 8th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI just mean: the functor category out of the abstract cellular $n$-cube $\Box^n$ regarded as a poset, regarded as a category. So for $n = 1$ the arrow category, for $n = 2$ the category of commutative squares, etc.

   I just mean: the functor category out of the abstract cellular $n$-cube $\Box^n$ regarded as a poset, regarded as a category. So for $n = 1$ the arrow category, for $n = 2$ the category of commutative squares, etc.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 10th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of the [phased tensor product](http://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+of+correspondences#PhasedTensorProduct) and expanded the [discussion of full dualizability](http://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+of+correspondences#FullDualizability) accordingly.

   added statement of the phased tensor product and expanded the discussion of full dualizability accordingly.