Timeline for Natural isomorphisms: what is the status now of "the Eilenberg/Mac Lane Thesis"?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8, 2015 at 9:22 | answer | added | Simon Henry | timeline score: 6 | |
| Oct 8, 2015 at 8:47 | comment | added | Fernando Muro | The lack of functoriality of the otherwise extremely natural notion of center is a source of counterexamples for all your questions. | |
| Sep 8, 2015 at 8:33 | answer | added | user80028 | timeline score: 4 | |
| Aug 12, 2015 at 20:58 | comment | added | Emily Riehl | I'm just writing to say I think this is a very interesting question. I don't find myself on mathOverflow that often, so if someone answers this, could somebody alert me? | |
| Jul 14, 2015 at 9:32 | comment | added | David Roberts♦ | @ZhenLin there's a bit of a difference, conceptually, between the case when the domain of the functors is small and large. When the domain is large, one can use instead a a weak category: then natural with respect to isomorphisms is univalence in disguise, I think. | |
| Jul 14, 2015 at 9:30 | comment | added | Zhen Lin | Sometimes, though, it turns out that naturality with respect to isomorphisms is trivial; for instance, the only isomorphisms in $\mathbf{\Delta}$ are the identities. I suppose this is no different to by-accident natural transformations... | |
| Jul 14, 2015 at 9:29 | comment | added | David Roberts♦ | However, the definition of a 'canonical' morphism is a bit more vague, and possibly is encoded in the idea of being natural with respect to isomorphisms only. This is perhaps what you were after? See ncatlab.org/nlab/show/core-natural+transformation for discussion, and the MO question mathoverflow.net/questions/19644/… | |
| Jul 14, 2015 at 8:36 | comment | added | Peter Smith | Yes, I know that E & MacL were after a general story about natural transformations: I thought, rightly or wrong, it would be helpful here to narrow the question by focusing on isomorphisms. | |
| Jul 14, 2015 at 8:20 | comment | added | David Roberts♦ | The contrapositive is false: there are natural morphisms that are defined by first choosing some arbitrary information which turns out to be irrelevant (e.g. picking a basis for a vector space, and then the result is independent of the choice). Also, don't just think of natural isomorphisms, since the important examples E&MacL started from were not (maps between cohomology theories, for instance Cech for different covers) | |
| Jul 14, 2015 at 7:49 | comment | added | Eric Wofsey | There are plenty of "natural" constructions in mathematics that are not functorial (or natural) with respect to the usual notion of morphism of the objects involved (for a simple example, consider the construction that takes a set $X$ and give the set of $2$-element subsets of $X$). What is much more robust (and I would even say without exception) is that any "natural" construction is functorial/natural with respect to isomorphisms. | |
| Jul 14, 2015 at 7:36 | history | asked | Peter Smith | CC BY-SA 3.0 |