Timeline for What do people mean by "subcategory"?
Current License: CC BY-SA 2.5
6 events
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| May 13, 2010 at 16:34 | comment | added | Ben Wieland | It's interesting that you're the only person to defend the official definition (maybe implicitly TJF). The second paragraph helped me: homotopy theorists find subspaces helpful, even if they are "evil." There are two useful notions of subcategory, so it's bad that people use the same word. I think you're right that pseudomonic is what people should mean if they want a categorical notion, but I'm not convinced it is what they mean. | |
| May 13, 2010 at 16:21 | vote | accept | Ben Wieland | ||
| May 13, 2010 at 6:41 | comment | added | Tilman | @Mike: we can probably argue endlessly on this one, but if you take the geometric realization of a category, equivalences go to homotopy equivalences and not to homeomorphisms. So it's not just a vague analogy I'm referring to here, it's an honest-to-god functor -- although it's of course true that geometric realization is by no means a subcategory inclusion... | |
| May 13, 2010 at 6:20 | comment | added | Mike Shulman | Absolutely disagreed on the second paragraph! Requiring a categorical notion to be invariant under equivalence is really much more like requiring the notion of inclusion of topological spaces to be invariant under homeomorphism. Homotopy equivalence is a weaker notion of equivalence imposed on topological spaces when we use them as models for "homotopy types;" the "natural" notion of sameness for topological spaces is homeomorphism, just as the natural notion of sameness for categories is equivalence. | |
| May 12, 2010 at 22:35 | comment | added | Peter LeFanu Lumsdaine | Absolutely agreed on the second paragraph! An example here of an interesting subcategory inclusion that is not full on isomorphisms: "differential manifolds; smooth maps" inside "differential manifolds; continuous maps". On the other hand, "pseudomonic" isn't the only notion of subcategory that's invariant under equivalence --- "faithful" is fine for that too, as are the other candidates discussed on the nlab at Andrew S's link. | |
| May 12, 2010 at 21:34 | history | answered | Tilman | CC BY-SA 2.5 |