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I would appreciate others views on the use of 'identify' in proofs. I was reading a paper recently and, having shown that a construction was independent up to isomorphism of the choices made, the author promptly said that they would identify the images of the construction thereby giving a functor. Although I must admit to using this sort of wording myself sometimes, my hackles went up. This seems a dangerous activity, identifying not raising hackles, that is. (Of course in the n-lab it is more or less evil if I understand `evil')
Does someone have a good example of when it really does miss the point, and how should one handle such situations?
In principle, identifying is bad, but for practical purposes, it really is necessary especially dealing with things like subrings and subgroups etc. (I once wrote up a proof of the finite case of the pontryagin duality without identifying injections with subgroups. It made the proof literally three times as long as it should have been.)
Exactly for subobjects this may be sometimes dangerous. There are categories with examples of an object which has a proper class of nonisomorphic subobjects. But worse, some definitins, like the Rosenberg's definition of a spectrum of an abelian category simulteneously involve parts of the definition which involve classes and part which involve equality. This is in that case subtle and I have seen experts in ring theory and algebraic geometry failing to understand it just because they took one of the stands instead of literally reading the definition. I hope to soon write an entry on this, but I am trying outside of the lab doing some connection of math which I am doing to physical applications and when I come back to nlab I get distracted badly back to purepurepure math. What happened with original intentions of nlab community ? Except for discussion of TQFTs almost no physics content so far.
They came for the physics but stayed for the higher categories, I assume.
Not being a physicist myself, I never had any intention of writing about physics. (-: I can't speak for anyone else.
I'm also not sure why identifying isomorphic subobjects would be a problem simply because there are a proper class of them.
What happened with original intentions of nlab community ? Except for discussion of TQFTs almost no physics content so far.
I intend to write more on physics. Have been intending this all along. All abstract math I write is motivated by the physics that I intend to write.
For instance I intend to eventually write a really correct POV discussion of BV-BRST. That requires derived stacks. The BV-BRST complex is precisely the global sections dg-algebra of a derived stack. It is precisely the construction in the first part of page 23 of the latest Ben-Zvi/Nadler:
there is an adjunction
between oo-sheaves on cochain dg-algebras in non-positive degree on the one hand and unbounded dg-cochain algebras on the other. The images under of derived oo-stacks are BV-BRST complexes: the stuff in positive degrees comes from the categorical degrees of the stack, the stuff in negative degrees from the negative degrees of the representable dg-algebras.
Just an example where I need to get lots of pure math into place to say what I would like to say about physics.
Apart from that it so happens that I am employed in a math department, not in a physics department. That has happened that way because the physics that I find interesting is done these days not in the physics departments. But the drawback is that I am forced to do more pure math than I would do if left entirely to my own desires.
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