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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJun 10th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCan anyone suggest a naturally-occurring monoidal category whose terminal object $1$ is not the unit of the monoidal structure and where also $1\otimes 1\ncong 1$?

   Can anyone suggest a naturally-occurring monoidal category whose terminal object $1$ is not the unit of the monoidal structure and where also $1\otimes 1\ncong 1$?

2. • CommentRowNumber2.
   • CommentAuthorOscar_Cunningham
   • CommentTimeJun 10th 2016
   • (edited Jun 11th 2016)
   • PermaLink
   Author: Oscar_Cunningham
   Format: MarkdownItexWhat about $\mathbf{Set}$, taking $\otimes$ to be the coproduct?

   What about $\mathbf{Set}$, taking $\otimes$ to be the coproduct?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 11th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexOkay, yeah, of course, I simplified too much. How about a *closed* monoidal category with these properties?

   Okay, yeah, of course, I simplified too much. How about a closed monoidal category with these properties?

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 11th 2016
   • (edited Jun 11th 2016)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexA cheap response is to tweak Oscar's example to $(Fin, \otimes = +)$, and pass to $Set^{Fin^{op}}$ via Day convolution. Why do you want to know? :-)

   A cheap response is to tweak Oscar’s example to $(Fin, \otimes = +)$, and pass to $Set^{Fin^{op}}$ via Day convolution.

   Why do you want to know? :-)

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 11th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexHere might be a "more natural" example: do the same as in the last comment, but with $\Delta$ in place of $Fin$, with its usual tensor product (corresponding to simplicial join). I.e., pass to simplicial sets with the induced tensor product.

   Here might be a “more natural” example: do the same as in the last comment, but with $\Delta$ in place of $Fin$, with its usual tensor product (corresponding to simplicial join). I.e., pass to simplicial sets with the induced tensor product.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJun 11th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAh, good one: the join on augmented simplicial sets. I'm thinking about substructural type theories for categories, such as a term calculus for intuitionistic linear logic (with $\otimes,1,+,0,\times,\top,\multimap$) that presents a free closed monoidal category with products and coproducts. And I noticed that the term for the unique inhabitant of $\top$ (the additive truth, i.e. the terminal object) can't be just "$tt$" the way it is in cartesian type theory, but has to take all the variables from the context as arguments. Otherwise a term like $x:\top \vdash \langle tt,tt\rangle : \top\otimes \top$ is ambiguous; we ought to write either $\langle \tt(x),tt\rangle$ or $\langle \tt,tt(x)\rangle$. But then I couldn't think of any examples of a naturally-ocurring monoidal category where this difference actually mattered. (Why am I thinking about this, you may ask? Because I'm preparing to [teach a short course](https://aarms.math.ca/the-2016-aarms-summer-school/) on categorical logic.)

   Ah, good one: the join on augmented simplicial sets.

   I’m thinking about substructural type theories for categories, such as a term calculus for intuitionistic linear logic (with $\otimes,1,+,0,\times,\top,\multimap$) that presents a free closed monoidal category with products and coproducts. And I noticed that the term for the unique inhabitant of $\top$ (the additive truth, i.e. the terminal object) can’t be just “$tt$” the way it is in cartesian type theory, but has to take all the variables from the context as arguments. Otherwise a term like $x:\top \vdash \langle tt,tt\rangle : \top\otimes \top$ is ambiguous; we ought to write either $\langle \tt(x),tt\rangle$ or $\langle \tt,tt(x)\rangle$. But then I couldn’t think of any examples of a naturally-ocurring monoidal category where this difference actually mattered.

   (Why am I thinking about this, you may ask? Because I’m preparing to teach a short course on categorical logic.)

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 11th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexAre you typing up notes to go with this? I'd enjoy seeing them.

   Are you typing up notes to go with this? I’d enjoy seeing them.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJun 13th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes; in fact the notes seem to be turning into something quite extensive, and I haven't even gotten to first-order logic yet. [Here](https://github.com/mikeshulman/catlog). I would welcome advance feedback, especially from category theorists on their comprehensibility, and from type theorists on their correctness.

   Yes; in fact the notes seem to be turning into something quite extensive, and I haven’t even gotten to first-order logic yet. Here. I would welcome advance feedback, especially from category theorists on their comprehensibility, and from type theorists on their correctness.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 13th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the pointer. Hm, on the github-page it says > The file you want to compile is I'd rather not compile any of your files, but just open a pdf.

   Thanks for the pointer. Hm, on the github-page it says

   The file you want to compile is

   I’d rather not compile any of your files, but just open a pdf.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 13th 2016
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexNot being the greatest tex expert, my compiler is struggling at line 4 (\newif\ifcref\creftrue), and I don't know what to do.

    Not being the greatest tex expert, my compiler is struggling at line 4 (\newif\ifcref\creftrue), and I don’t know what to do.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 13th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexAlso, unless one also grabs the bibtex file, none of the references will work (I cloned the repo in my Desktop installation of GitHub and ran 'build' in Sublime Text, which took some time but still gave lots of warnings. I didn't bother trying to get the refs)

    Also, unless one also grabs the bibtex file, none of the references will work (I cloned the repo in my Desktop installation of GitHub and ran ’build’ in Sublime Text, which took some time but still gave lots of warnings. I didn’t bother trying to get the refs)

12. • CommentRowNumber12.
    • CommentAuthorMatt Earnshaw
    • CommentTimeJun 13th 2016
    • (edited Jun 13th 2016)
    • PermaLink
    Author: Matt Earnshaw
    Format: MarkdownItex[here's a pdf](https://github.com/mattearnshaw/catlog/raw/master/catlog.pdf) for the impatient :) (edit: added refs)

    here’s a pdf for the impatient :) (edit: added refs)

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2016
    • PermaLink
    Author: Urs
    Format: MarkdownItexp. 11 in "exhibit the most behavior most characteristic of type theories" probably one superfluous "most"

    p. 11 in “exhibit the most behavior most characteristic of type theories” probably one superfluous “most”

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJun 13th 2016
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexDavid C, I can't help you unless you can be more specific than "struggling". My bib file is in [this repo](https://github.com/mikeshulman/basictex). Sorry everyone, this is a working draft and I don't have the time to distribute pretty copies myself. (-:

    David C, I can’t help you unless you can be more specific than “struggling”. My bib file is in this repo. Sorry everyone, this is a working draft and I don’t have the time to distribute pretty copies myself. (-:

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2016
    • (edited Jun 14th 2016)
    • PermaLink
    Author: Urs
    Format: MarkdownItexSo with all this repository technology, there is no way that whenever you recompile your pdf, the output gets uploaded to the git-hub page?

    So with all this repository technology, there is no way that whenever you recompile your pdf, the output gets uploaded to the git-hub page?

16. • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 14th 2016
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI can see Matt's (#12), sot that's fine.

    I can see Matt’s (#12), sot that’s fine.

17. • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJun 14th 2016
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexThe whole point of source control software is to control the *source*, not the compiled version. Is it really so hard to git clone?

    The whole point of source control software is to control the source, not the compiled version. Is it really so hard to git clone?

18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeJul 10th 2016
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexFYI, [here](http://mikeshulman.github.io/catlog/catlog.pdf) is a compiled current version that I'm going to point the students at tomorrow.

    FYI, here is a compiled current version that I’m going to point the students at tomorrow.