Timeline for A category with objects that are not based on sets or classes
Current License: CC BY-SA 3.0
5 events
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| Dec 17, 2012 at 23:11 | comment | added | David Roberts♦ | DOOOOOOOOMED!!!11! - even if you don't demand local smallness! (which Tim is assuming) Because there might be a class of arrows between two objects! | |
| Dec 17, 2012 at 18:20 | comment | added | Tim Porter |
As someone else pointed out in the usual DEFINITION of a category you are required to have a SET of morphisms between objects, so from that point of view, your search is DOOMED! If you take a different foundation' for mathematics, what one do you want? You can follow Lawvere's idea of using categories as the basic things from which to build things... note this is not a foundational exercise as such, rather a pragmatic one. Here is an idea: take categories as basic, then the 2-category of categories (no size to be mentioned since set theory is anathema') and functors might pass must. :-)
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| Dec 17, 2012 at 18:12 | history | edited | Tim Porter | CC BY-SA 3.0 |
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| Dec 17, 2012 at 12:53 | comment | added | jd94j39 | Thanks Tim for the answer, but to me- all these things you have mentioned are constructions from sets. The "elements" of the group are morphisms in a category with one point. But these morphisms form a set. In the second example, the objects are points of a topological space, and a topological space is constructed from sets. I can say the same thing for the third example. I know that I am not being precise with what I am asking, but it is difficult if sets seem to loom over everything I see in mathematics. | |
| Dec 17, 2012 at 12:46 | history | answered | Tim Porter | CC BY-SA 3.0 |