Timeline for Constructions that can be seen as objects representing a functor
Current License: CC BY-SA 4.0
14 events
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| Sep 13, 2019 at 3:58 | comment | added | Praphulla Koushik | I see a downvote... any suggestion on improving the question? | |
| Sep 1, 2019 at 16:30 | review | Close votes | |||
| Sep 9, 2019 at 3:05 | |||||
| Sep 1, 2019 at 12:30 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Aug 31, 2019 at 16:58 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
added 248 characters in body
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| Aug 31, 2019 at 16:56 | comment | added | Praphulla Koushik | @DmitriPavlov Yes, I should have taken homotopy classes in first case.. In third case, I mean homotopy class of maps, I did not mention, should have mentioned... I am looking for some constructions which are not obvious cases of limits, colimits etc. Does this question then have some non trivial answer? | |
| Aug 31, 2019 at 16:30 | comment | added | Dmitri Pavlov | Concerning the general question: almost all definitions in category theory can be expressed using representable functors, including limits (such as products, pullback, equalizers), colimits (such as coproducts, pushouts, coequalizers), left and right adjoint functors, ends and coends, left and right Kan extensions, etc. So the question, quite literally, asks to list a substantial fraction of mathematics. | |
| Aug 31, 2019 at 16:27 | comment | added | Dmitri Pavlov | The first and third examples are incorrect as stated. Top is a category, so the set of morphisms M→F_G cannot be equivalent (let alone isomorphic) to a groupoid. The third example suffers from a similar mistake: one must take homotopy classes of maps, so this will not be a representable functor on Top. | |
| Aug 31, 2019 at 12:04 | comment | added | Praphulla Koushik | @vidyarthi Oh...So, the functor is the forgetful functor.. I will try to see if this is representable and which objects represents it... | |
| Aug 31, 2019 at 11:47 | comment | added | vidyarthi | @PraphullaKoushik I meant the functor is from the category of graphs to the category of the sets defined by the graph being mapped to its set of homomorphisms/ group of homomorphisms. By "consider the category of quivers", I meant to consider the the free category of quivers instead of the category of graphs | |
| Aug 31, 2019 at 11:38 | comment | added | Praphulla Koushik | @vidyarthi what does it mean to say "functor consisting of graph morphisms" or "consider the category of quivers"? what exactly is the functor, from where to where? what object represents that functor? | |
| Aug 31, 2019 at 11:37 | comment | added | Praphulla Koushik | @LaurentMoret-Bailly I am not able to fill details immediately but I do agree there is a close relation between "universal property" and representable functors... Thank you for the comment.. I will try to make my question more sensible... | |
| Aug 31, 2019 at 10:36 | comment | added | vidyarthi | A probable example would be the finite category of graphs and the functor consisting of graph homomorphisms. May be it would be representable. Or consider the category of quivers | |
| Aug 31, 2019 at 9:23 | comment | added | Laurent Moret-Bailly | Any "universal property" can be expressed in terms of a representable functor, so I wouldn't know where to start, or where to stop. | |
| Aug 31, 2019 at 7:01 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |