Timeline for Constructing unnatural transformations
Current License: CC BY-SA 3.0
21 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Apr 11, 2015 at 11:10 | comment | added | Karol Szumiło | The most natural example (I can think of) of a category where morphisms ignore much of the structure would be the category of metric spaces and continuous maps. Without being aware of the concept of topological spaces we wouldn't be able to recognize how much structure is ignored here. How could we ever be confident that something like that doesn't happen in other examples? | |
| Apr 11, 2015 at 3:35 | comment | added | Eric Wofsey | I would also note that in the usual framework of material set theory, $Set$ itself is an example of a category in which the objects have extraneous data. For instance, here's an extremely unnatural transformation on $Set$: let $F:Set\to Set$ be the constant functor to some singleton set and let $G:Set\to Set$ be the covariant power set functor. Let $\mu_x:F(x)\to G(x)$ be the transformation that picks out the set of all elements of $x$ which are also elements of elements of $x$. | |
| Apr 11, 2015 at 3:15 | comment | added | Eric Wofsey | More generally, you can easily come up with examples along the lines of Karol's whenever the objects of your category contain "more data" than the morphisms can detect. While Karol's example is artificial, non-artificial examples are not uncommon in actual mathematical practice. For instance, many people define a "smooth manifold" as a topological space equipped with a (not necessarily maximal) smooth atlas. While you can argue that this definition is "wrong", it seems very difficult to find a precise way to exclude such definitions for the purposes of your question. | |
| Apr 11, 2015 at 1:23 | answer | added | user70198 | timeline score: 9 | |
| Apr 10, 2015 at 21:54 | review | Close votes | |||
| Apr 11, 2015 at 7:27 | |||||
| Apr 10, 2015 at 18:10 | comment | added | Karol Szumiło | I figure that my example is "cheated" and not what you want, but my point is that you haven't specified how exactly you want to prohibit an example like that. In particular, I don't understand your objection that "the categories are different". The way you posed your question I am free to choose the categories how I want. And I chose $\mathsf{Aff}'$ rather than $\mathsf{Aff}$. | |
| Apr 10, 2015 at 17:36 | comment | added | Anton Fetisov | @ZhenLin, not sure how you modify it. Do you mean two constant functors $c \mapsto 1$ and $c \mapsto C $ with transformation $c \mapsto c: 1 \to C $? Because it is also lax natural: $f: c \to c'$ gives 2-cell $f: c \to c'$. | |
| Apr 10, 2015 at 17:18 | comment | added | Zhen Lin | That is true, but I could easily modify the example by replacing $C_{/ c}$ with $C$ itself. Now there is no room to even insert a mediating 2-cell. | |
| Apr 10, 2015 at 17:11 | comment | added | Anton Fetisov | @ZhenLin, nice example, but it's natural in the naturally generalized sense. For functors into $Cat $ we should consider lax naturality, and for both $F: C \to Cat, Fc := C/_c $ and $F: C^{op} \to Cat, Fc := C/_c $ the transformation $1 \to F $ is lax natural (even pseudo natural in the second case) and represented by $1 \to Cat $. | |
| Apr 10, 2015 at 16:58 | comment | added | Joel David Hamkins | My point was that you will not be able to prove that definable maps are natural, because there are models of set theory in which ALL maps (of any kind, from any set or class to another), including those coming from AC, are definable. | |
| Apr 10, 2015 at 16:53 | comment | added | Anton Fetisov | @JoelDavidHamkins, AC is manifestly non-functorial. Specifically, I have in mind HoTT, where we can prove that AC for anything more than 0-types is inconsistent. It is not surprising that a non-functorial axiom allows us to define non-functorial families. Also your statement is specific to Godel's model, while I'm interested in model-independent results. | |
| Apr 10, 2015 at 16:45 | comment | added | Joel David Hamkins | You seem in your question to presume that uses of AC are inherently non-definable. But this is not a correct intuition. In fact, the way that we first came to know that AC is consistent is through Godel's constructible universe $L$, where there is a definable well-ordering of the universe. Thus, one can use AC in this universe and still have a definable function. Worse, there are other models of ZFC and even GBC in which every set and class is definable without parameters. So using AC in such a model cannot produce a non-definable class. My view is that the question has a problem at its core. | |
| Apr 10, 2015 at 16:42 | comment | added | Zhen Lin | Here's a natural example of an "unnatural" morphism: for each object $c$ in $C$, there is a functor $1 \to C_{/ c}$ picking out the object $(c, \mathrm{id}_c)$. $C_{/ c}$ is functorial in $c$ in an obvious way, and $1$ is just constant – so naturality amounts to saying that this defines a cone over the diagram $C_{/ \bullet}$, but it does not. | |
| Apr 10, 2015 at 16:27 | comment | added | David Roberts♦ | @AsafKaragila, no, more like this: goo.gl/pWM9GI | |
| Apr 10, 2015 at 16:20 | comment | added | Anton Fetisov | @KarolSzumiło, my goal is not to devise some devious way to equate the non-equatable, but rather to understand what can happen when one is restricted to natural constructions. Can we quantify the unnaturality of results? | |
| Apr 10, 2015 at 16:17 | comment | added | Anton Fetisov | @KarolSzumiło, it's not exactly a counterexample to my point since the categories are different. In the case of $Set$ your construction would take the category of well-ordered sets with all maps. Since the equivalence requires choice, it doesn't actually help you for functors on $Set$, but the point is valid: in general it is not so clear which categories $C$ are good enough. I'd say that $C$ should be $Set$ or explicitly definable over $Set$. | |
| Apr 10, 2015 at 15:58 | comment | added | Asaf Karagila♦ | Is it customary, after constructing an unnatural transformation to shout in a trembling voice "It's alive! ALIVE!!!" or do you guys don't do that? | |
| Apr 10, 2015 at 15:51 | comment | added | Karol Szumiło | AC is still needed to write down an equivalence $\mathsf{Aff} \to \mathsf{Aff}'$ but I don't think this contradicts any of your requirements. | |
| Apr 10, 2015 at 15:51 | comment | added | Karol Szumiło | I have trouble seeing how this is a well-posed question. You could always artificially avoid using the axiom of choice by building the choices into the structure of objects but making the morphisms ignore that extra structure. To be more specific, you could modify the example with affine spaces by taking $\mathsf{Aff}'$ to be category whose objects are affine spaces with a basepoint and morphisms are just affine (not necessarily based) maps. Then the definition of an unnatural transformation doesn't require AC. | |
| Apr 10, 2015 at 15:09 | history | asked | Anton Fetisov | CC BY-SA 3.0 |