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This is inspired by The Whitehead for maps question.

Consider two maps f, g: X\to Y which happen to induce the same maps (of discrete spaces) [Z, X] \to [Z, Y] for every Z. Does this mean f and g are homotopic?

And what would be the lessons from the answer to this question? I feel like there's something interesting about the way we should ask it.

flag	asked Oct 26 2009 at 21:34 Ilya Nikokoshev 9,019●5●30●73

Eric Wofsey · Accepted Answer · 2009-10-26 21:39:34Z UTC

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Yes, this is a special case of Yoneda. Let Z=X and consider the identity map in [X,X]; the hypothesis says that f1=f and g1=g are then equal as elements of [X,Y].

answered Oct 26 2009 at 21:39

Eric Wofsey
8,887●1●26●46

I didn't see your answer when I was composing mine for some reason :) – Ilya Nikokoshev Oct 26 2009 at 21:47

Ilya Nikokoshev · Answer 2 · 2009-10-26 21:43:02Z UTC

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I found it myself: the image of id \in [X, X] under both maps will be the same class in [X, Y], which is the definition of homotopy between f and g, so the ansewr is yes.

answered Oct 26 2009 at 21:43

Ilya Nikokoshev
9,019●5●30●73

Something like Yoneda’s lemma

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