Timeline for Natural isomorphisms: what is the status now of "the Eilenberg/Mac Lane Thesis"?
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Oct 8, 2015 at 10:23 | comment | added | sure | Yet, the internal natural transformations are not natural (as far as I know, even if you try to code them as modifications between 2-cats). So while the category of internal categories in C forms a 2-category, it seems that there is no functorial way to see them as a representation of a syntactic 2-category. Thus, the isomorphism of internal functors is not functorial, and therefore not natural. | |
| Oct 8, 2015 at 10:23 | comment | added | sure | I would like to support @user80028's view that the good question is about trying to understand what kind of construction is functorial. There is a clear example with 2-category theory: if C is a category, an internal category to C is just a left exact functor from the sketch of an (internal) category to C. Now, an internal functor between these categories is just a natural transformation between the previous functors. | |
| Oct 8, 2015 at 9:06 | comment | added | Fernando Muro | @SimonHenry nice observation which leads to another tautological answer to the original question, this time positive. Going a little bit further, if we remove non-identity morphisms from a given category, any construction becomes natural. | |
| Oct 8, 2015 at 8:47 | comment | added | Simon Henry | The two construction are functorial on isomorphisms and the isomorphism between then is a natural transformation in this sense. But of course this kind of general principle only apply to isomorphism... | |
| Sep 8, 2015 at 8:54 | review | Late answers | |||
| Sep 8, 2015 at 8:57 | |||||
| Sep 8, 2015 at 8:34 | review | First posts | |||
| Sep 8, 2015 at 8:57 | |||||
| Sep 8, 2015 at 8:33 | history | answered | user80028 | CC BY-SA 3.0 |