Timeline for Would it be possible to propose a satisfying categorical definition for the notion of basis?
Current License: CC BY-SA 4.0
6 events
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| Apr 25, 2023 at 20:23 | comment | added | Mike Shulman | I think whether a given functor is "forgetful" really does depend on signatures or presentations of the categories involved. For any functor at all, we can replace its domain by an equivalent categories so as to make it look forgetful. For instance, the category of sets is equivalent to the category of vector spaces equipped with a basis and linear maps that preserves these bases; the free functor from this category of "sets" to the category of vector spaces is then simply "forgetting the bases". | |
| Apr 23, 2023 at 19:02 | comment | added | Dave Benson | I guess I agree. Even MacLane uses the term in "Categories for the Working Mathematician" via examples. | |
| Apr 23, 2023 at 18:32 | comment | added | LSpice | Re, as far as I can tell, although wiki:forgetful functor gives examples of forgetful functors, and properties they "tend to" have, it doesn't actually define what a forgetful functor is, only what it does. More precisely, it suggests a definition via universal algebra ("curtailing the signature"), but I don't see how to use it to recognise when a functor between abstract categories (that I do not know a priori consists of, say, algebras of a given signature) is, or is not, forgetful. | |
| Apr 23, 2023 at 17:08 | comment | added | Dave Benson | I recommend the Wikipedia page on forgetful functors. | |
| Apr 23, 2023 at 16:49 | comment | added | LSpice | Is "forgetful functor" an informal term, or is there some formal way to define when a functor is forgetful? Or is it a datum that is to be specified together with the category? | |
| Apr 23, 2023 at 14:17 | history | answered | Dave Benson | CC BY-SA 4.0 |