Timeline for Do normal categories have pullbacks?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 6, 2010 at 16:38 | vote | accept | José Figueroa-O'Farrill | ||
| Sep 6, 2010 at 14:34 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 2.5 |
pointed to comments for explanation
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| Sep 6, 2010 at 14:32 | comment | added | Peter LeFanu Lumsdaine | @Martin: the “dimension argument” I had in mind was looking at the dimensions of hom-sets, not the objects themselves, which as you say wouldn't be a proof. But Todd's argument gives a nicer big picture to see it in, I think. | |
| Sep 6, 2010 at 12:40 | comment | added | Todd Trimble | ncatlab.org/nlab/show/reflected+limit | |
| Sep 6, 2010 at 12:37 | comment | added | Todd Trimble | I actually quite liked Peter's example. As for the objection, one might observe that the underlying functor from Peter's $Vect_n$ to $Set$ preserves any limits which exist in $Vect_n$ (since it is representable), and reflects them as well since it reflects isomorphisms (see the <a href="ncatlab.org/nlab/show/reflected+limit">nLab page</a>). So any product which exists in $Vect_n$ is the expected one. | |
| Sep 6, 2010 at 6:34 | comment | added | Martin Brandenburg | It is not obvious at all that your category has no products. [The products in the bigger category are not in it, but this is not a proof] | |
| Sep 6, 2010 at 2:10 | comment | added | José Figueroa-O'Farrill | Many thanks for this. Indeed it is not hard to show that a normal category has pullbacks of monos and this is what ends up being shown in the book. I would be happy to add products to the definition of normal, since this is what most of the other results depend on. Blyth gets this as a consequence of the existence of pullbacks (and the existence of a zero object). | |
| Sep 6, 2010 at 1:47 | history | answered | Peter LeFanu Lumsdaine | CC BY-SA 2.5 |