3 ms·
Any two of those many products, however, are isomorphic, and hence, while still distinguishable, they are “essentially” the same.
by mmarx 12y ago
Any two of those many products, however, are isomorphic, and hence, while still distinguishable, they are “essentially” the same.
- tel 12y agoIf I'm reading the op right, this is incorrect. Oftentimes "products" in various contexts are "categorical products cut down by some sensible quotienting". In any category the categorical product will always be the largest (this is essentially what its definition entails) and only by careful choice of category will that correspond to some of the less formal "products" available.
- mmarx 12y agoNo, altough the wording might have been misleading—I was refering to the second paragraph, while you seem to have applied it to the first. I hope this clarifies the situation a bit: For any category, the product A * B of two objects A, B (if it exists) is unique only up to isomorphism (this follows from the universal mapping property). In general, products A * B and C * D need not be isomorphic (and seldomly are), and neither do products from different categories. Many of the commonly-used products actually arise from a product in a suitable category, the cartesian product, for example, is a product in the category of sets.
- tel 12y agoYeah I think I just misunderstood your statement: I agree with this extension upon it completely.