Timeline for Equivalence of categeories-variants of definition [closed]
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| May 16, 2016 at 6:28 | review | Reopen votes | |||
| May 16, 2016 at 8:04 | |||||
| May 16, 2016 at 6:11 | history | edited | truebaran | CC BY-SA 3.0 |
added 283 characters in body; edited title
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| May 15, 2016 at 23:45 | history | closed |
Andrej Bauer abx Qiaochu Yuan Wolfgang Myshkin |
Needs details or clarity | |
| May 15, 2016 at 21:20 | comment | added | truebaran | David Roberts thank you, you are obviosly right, I corrected what was wrong | |
| May 15, 2016 at 21:18 | history | edited | truebaran | CC BY-SA 3.0 |
added 86 characters in body
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| May 15, 2016 at 20:52 | comment | added | David Roberts♦ | It is false that faithful means "injective on arrows". Please check the definition again, as the real definition of fully faithful doesn't imply injectivity on objects. Eg the functor M from the category of vector spaces over R with basis, and all linear maps, to the category with objects the natural numbers and matrices as morphisms: this is an equivalence of categories and highly non-injective on objects and arrows. Also, use 'injective up to isomorphism' becasue that is what you really mean, or 'essentially injective'. | |
| May 15, 2016 at 18:41 | vote | accept | truebaran | ||
| May 15, 2016 at 18:38 | comment | added | Andrej Bauer | Ok, not really a research-level question in my opinion, it would be better to ask this on math.stackexchange.com. | |
| May 15, 2016 at 18:38 | answer | added | Andrej Bauer | timeline score: 7 | |
| May 15, 2016 at 18:31 | review | Close votes | |||
| May 15, 2016 at 23:45 | |||||
| May 15, 2016 at 18:26 | comment | added | truebaran | ... this I meant to be implemented by $F$, | |
| May 15, 2016 at 18:26 | comment | added | truebaran | Not implemented by $F$ means that for each pair of objects $C_1,C_2$ there is a map $T_{C_1,C_2}$ between sets $Hom_{mathcal{C}}(C_1,C_2)$ and $Hom_{\mathcal{D}}(F(C_1),F(C_2))$ which is a bijection. This map sends a morphism $f:C_1 \to C_2$ to some morfizm between $F(C_1) \to F(C_2)$ in such a way that $T_{C_1,C_2}f_1=T_{C_1,C_2}f_2$ implies $f_1=f_2$ and each morphism in $Hom_{\mathcal{D}}(F(C_1),F(C_2))$ is of the form $T_{C_1,C_2}(f)$ for some morphism $f \in Hom_{\mathcal{C}}(C_1,C_2)$. In the definition of equivalence of categories $T_{C_1,C_2}(f)$ is just $F(f)$... | |
| May 15, 2016 at 18:14 | comment | added | Andrej Bauer | What is an "abstract" bijection and what does it mean for "a bijection to be implemented by $F$"? | |
| May 15, 2016 at 18:05 | history | asked | truebaran | CC BY-SA 3.0 |