Timeline for Are there noncartesian monoidal categories with $A \otimes B = A \times B$?
Current License: CC BY-SA 4.0
6 events
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| Jul 27 at 7:58 | comment | added | John Baez | Since the question asked for an example where $A \otimes B = A \times B$ for a chosen product $A \times B$, instead of merely $A \otimes B \cong A \times B$, it's good that you restricted to a 2-object category where this equation is indeed inevitable. Of course this demand for an equation rather than an isomorphism is purely for educational purposes, to drive the point home more strongly: any true category theorist will consider this extra demand a bit silly. But it's easy to satisfy this extra demand. | |
| Jul 26 at 9:26 | comment | added | Peter LeFanu Lumsdaine | @NaïmFavier: On the whole of $\mathrm{Set}_\bullet$, product and coproduct don’t coincide — they differ on non-unit finite sets. But yes, we don’t have to restrict as much as I did — we could take all unit-or-countably-infinite pointed sets; or (assuming choice) all unit-or-infinite ones. I chose this very small example to give a case where it’s easy to make the cartesian and non-cartesian structures literally equal on objects. | |
| Jul 26 at 9:26 | comment | added | John Baez | @NaïmFavier - The coproduct of pointed sets is generally not isomorphic to their product, so we must restrict to objects for which their product is isomorphic to their coproduct. Choosing just two, all 4 possible products of these objects are actually equal to the corresponding coproducts. | |
| Jul 26 at 9:22 | comment | added | John Baez | Excellent! Like my own conjectured solution (see original question) you are taking a cartesian category with a semicartesian monoidal structure and restricting to the full subcategory consisting just the unit object and one object $x$ with $x \times x \cong x \otimes x \cong x$. The advantage is that you're working with a much simpler category, and you've got a great way to show the semicartesian monoidal structure doesn't become cartesian when restricted to this subcategory. Efficient! | |
| Jul 26 at 9:21 | comment | added | Naïm Favier | Sorry, what's the point of the restriction? Doesn't this all work in pointed sets? | |
| Jul 26 at 9:12 | history | answered | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |