Timeline for Global elements in categories with no terminal object?
Current License: CC BY-SA 3.0
11 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Sep 12, 2016 at 8:55 | comment | added | HeinrichD | Notice that this notion of a global element very much looks like the definition of a global section, glued from local sections, of a sheaf. | |
| Jul 8, 2016 at 3:24 | comment | added | Mike Shulman | There really are two different notions of "constant function" even for plain old sets: according to one (factoring through the terminal object) the identity function of the empty set is not constant, while according to the other (the images of any two elements of the domain are equal) it is. See for instance ncatlab.org/nlab/show/constant+morphism, although the second definition there currently needs some fixing (nforum.ncatlab.org/discussion/2689/constant-morphism/…) | |
| Jul 6, 2016 at 23:17 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 6, 2016 at 23:15 | comment | added | Qiaochu Yuan | @Mike: I think you're right, except that if $C$ has a terminal object, then by the fully faithfulness (?) of the Yoneda embedding I still think this definition should reduce to the usual one. What am I missing? (As for your example, by either definition the empty set fails to have a global element and so no function to the empty set is constant.) | |
| Jul 6, 2016 at 15:25 | comment | added | Mike Shulman | I think your "unwound" definition of constant morphism is not exactly what you get by unwinding the definition in terms of the terminal presheaf. The latter would say "we can choose for each object $x$ a morphism $f_x : x \to d$ such that for any $g:y\to x$ we have $f_x g = f_y$ and moreover for any $h:x\to c$ we have $f h = f_x$." They are equivalent if every homset $\mathrm{Hom}(x,c)$ is nonempty, but otherwise the definition using the terminal presheaf is stronger (even if the category does have a terminal object). For instance, consider the identity function on the empty set. | |
| Jul 5, 2016 at 18:41 | comment | added | Qiaochu Yuan | @Marcus: it seems like a pretty natural construction to me, but I don't know a reference for it. | |
| Jul 5, 2016 at 12:13 | vote | accept | Marcus Pivato | ||
| Jul 5, 2016 at 12:11 | comment | added | Marcus Pivato | Thank you very much, Qiaochu! This is exactly what I was looking for. In fact, I already had in mind something like your second grey box (i.e. below "Unwinding these definitions, we get the following.") But I had not made the connection to the Yoneda embedding. (I am not an expert in category theory.) ----- My remaining question: is this a standard construction, or is it something you just invented? If it is a standard construction, then could you point me to some literature where this construction is defined and its properties are developed? Thanks again! | |
| Jul 4, 2016 at 19:56 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 4, 2016 at 19:21 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |