Timeline for Two kinds of equivalence: conjugate vs. isomorphic objects
Current License: CC BY-SA 4.0
16 events
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| Oct 11, 2018 at 17:30 | comment | added | R. van Dobben de Bruyn | It should be true that isomorphic objects are always conjugate. Given an isomorphism $f \colon X \stackrel \sim \to Y$ in $\mathscr C$, you can construct the endofunctor $F \colon \mathscr C \to \mathscr C$ that is the identity on all objects except $X$ and $Y$, but swaps $X$ and $Y$. On morphisms, it is the identity on $\operatorname{Hom}(A,B)$ if $A, B \not \in \{X,Y\}$, and for $B \in \{X, Y\}$ we postcompose with $f$ or $f^{-1}$, and for $A \in \{X, Y\}$ we precompose with $f$ or $f^{-1}$. Then $F$ is an auto-equivalence, and in fact is isomorphic to the identity functor. | |
| Oct 11, 2018 at 11:13 | history | edited | Hans-Peter Stricker | CC BY-SA 4.0 |
added 68 characters in body
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| S Jan 2, 2018 at 15:08 | history | suggested | jeq | CC BY-SA 3.0 |
Copied images to imgur.com, as they were not being displayed because of new https rule. Added links to original image sources.
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| Jan 2, 2018 at 14:11 | review | Suggested edits | |||
| S Jan 2, 2018 at 15:08 | |||||
| Nov 3, 2011 at 2:35 | answer | added | Will Sawin | timeline score: 4 | |
| Nov 2, 2011 at 20:44 | answer | added | Karol Szumiło | timeline score: 5 | |
| Nov 2, 2011 at 18:06 | comment | added | Theo Johnson-Freyd | +1 for appendices and footnotes. | |
| Nov 2, 2011 at 15:49 | comment | added | Finn Lawler | @Hans: the former -- [A is conjugate to B in the category X] iff [(A,X) is isomorphic to (B,X) in */Cat]. | |
| Nov 2, 2011 at 15:23 | comment | added | Hans-Peter Stricker | @Finn: How do I have to phrase your answer? (A and B are conjugate) if and only if (they are isomorphic in the category of pointed categories) or (A and B are conjugate if and only if they are isomorphic) in the category of pointed categories | |
| Nov 2, 2011 at 13:06 | comment | added | Guillaume Brunerie | In the category of graphs there are two morphisms, sending the two vertices to one vertex. But if you are working in the category of graphs over the discrete graph with two objects, I agree that there are no morphisms. | |
| Nov 2, 2011 at 13:04 | comment | added | Finn Lawler | The two notions are not unconnnected -- A and B are conjugate if and only if they are isomorphic in the category of pointed categories (i.e. categories equipped with a distinguished object, with morphisms functors that strictly preserve these). | |
| Nov 2, 2011 at 13:00 | comment | added | Hans-Peter Stricker | Which morphisms did you mean (in which category)? | |
| Nov 2, 2011 at 12:47 | comment | added | Guillaume Brunerie | Oh, sorry, if you are in the category of graphs over two fixed vertices, there are indeed no morphisms, I misread this part. | |
| Nov 2, 2011 at 12:43 | comment | added | Guillaume Brunerie | Also, there are (two) morphisms between your right graph and your left graph. | |
| Nov 2, 2011 at 12:40 | comment | added | Guillaume Brunerie | There is a simpler example of conjugate object which are not isomorphic, take any two different objects in a discrete category. | |
| Nov 2, 2011 at 12:19 | history | asked | Hans-Peter Stricker | CC BY-SA 3.0 |