Timeline for Concise definition of subobjects
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2017 at 5:07 | comment | added | Mike Shulman | Secondly, the whole point of HoTT is that it syntactically enforces homotopy invariance, and thus in particular is unable to distinguish internally between discrete and homotopy-discrete groupoids. | |
| Jul 15, 2017 at 5:06 | comment | added | Mike Shulman | @AntonFetisov I think your last sentence is wrong in two ways. Firstly, distinguishing discrete groupoids from homotopy discrete groupoids has a long history and many applications. For instance, the coherence theorem of monoidal categories says that certain groupoids are homotopy discrete; but they are certainly not discrete. And the Barratt-Eccles operad is the nerve of an operad in Gpd consisting of homotopy-discrete (indeed, contractible) but not discrete groupoids; if they were replaced by discrete ones then the whole point would be lost. | |
| Oct 13, 2014 at 19:36 | vote | accept | Martin Brandenburg | ||
| Oct 11, 2014 at 22:22 | comment | added | Anton Fetisov | For sets subobjects should equal subsets, and while a subset $U \subset S$ can be embedded differently (in $#Aut(U)$ ways), it doesn't make sense to distinguish these embeddings. Even if you do, your data will just split into automorphisms of subobject (in any category) and the standard category of subojects, so a simplification seems reasonable. Distinguishing groupoids that are literally discrete from homotopy equivalent to discrete ones is a matter of taste which no one cared about before HoTT came. | |
| Oct 11, 2014 at 22:14 | review | Close votes | |||
| Oct 12, 2014 at 6:32 | |||||
| Oct 11, 2014 at 22:10 | comment | added | Martin Brandenburg | Sure, we lose nothing, but this was not my question ... in my opinion, a definition should capture the essential idea behind a concept, and I am not convinced that I should identify isomorphic monomorphisms. | |
| Oct 11, 2014 at 22:03 | comment | added | Anton Fetisov | I assume the only real reason is that the concept of subobject was invented long before any categorical riff-raff, especially long before higher category theory. At the time it seemed especially perverse to consider any functor representable besides a functor into $Set$. From a modern viewpoint you lose nothing, since the groupoid of monomorphisms is discrete. | |
| Oct 11, 2014 at 21:58 | answer | added | Todd Trimble | timeline score: 8 | |
| Oct 11, 2014 at 21:53 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Oct 11, 2014 at 21:52 | comment | added | Martin Brandenburg | Yes, I know that. The process basically takes a preorder and makes it a partial order. I think people like partial orders more than preorders - is this the only reason? | |
| Oct 11, 2014 at 21:50 | comment | added | მამუკა ჯიბლაძე | Well still another principle on the next level is that just as in a category the correct notion is isomorphism of objects, not equality, in a 2-category the correct notion is equivalence of objects, not isomorphism. Now subobjects of an object $A$ in a category in your sense themselves form a category, i. e. an object of the 2-category of categories. Subobjects in the old sense form another category. And these two categories are equivalent. So... | |
| Oct 11, 2014 at 21:47 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Oct 11, 2014 at 21:40 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |