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Sascha Haupt (Aug 23 2024 at 15:13):As someone who is learning about monoidal categories, and who has read some basic texts about closed cartesian categories and topos theory before, the following question came to my mind:
There is a monoidal structure induced on Set by the cartesian product. A topos has all finite limits, in particular it has all binary products. Do binary products induce a monoidal structure on an arbitrary topos? If they don't (I somehow suspect that), what is an easy example for a topos without a monoidal structure imposed by binary products?
Jean-Baptiste Vienney (Aug 23 2024 at 15:20):Every category with binary products and a terminal object can be equipped with a monoidal structure so you will not find a topos which can't be equipped with such a monoidal structure.
Ralph Sarkis (Aug 23 2024 at 16:08):The definition of a monoidal category is self-dual, so this also works with coproducts and initial objects.
John Baez (Aug 23 2024 at 16:16):And every topos has coproducts and an initial object, so that's a second way you can make any topos into a monoidal category.
Morgan Rogers (he/him) (Aug 24 2024 at 11:09):If you're interested in monoidal structures on toposes more generally, you might want be interested in [[Day convolution]].
John Baez (Aug 24 2024 at 14:07):That's a great way to get monoidal structures that are neither cartesian nor cocartesian.