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Jim, please help me, how can it be “very misleading” if it is “relevant for mapping to or from it”?
If an object that looks like X needs to be regarded as Y for understanding its morphisms, then I think it makes thinks clearer to think of it as a Y.
But I don’t want to fight about such terminology issues. I have changed the title of the entry. The text already commented on this issue before.
I did define $end(V)$ as a dg-Lie algebra. But I called the entry “endomorphism $\infty$-Lie algebra” because I think what is most important about this dg-Lie algebra is to know that it is a model for the endomorphism Lie algebra in an $\infty$-context.
Around here we routinely say “Lie 2-algebra” for something coming from a differential crossed module. But a differential crossed module is just a certain dg-Lie algebra. Nevertheless, calling it not a crossed module and not a dg-Lie algebra but a Lie 2-algebra is good: it reminds us that the concrete implementation of this gadget as this or that is not so important, but that what is important is its meaning as a higher Lie algebra.
In this spirit I thought (and still think, to be frank) that there ought to be an entry called “endomorphism $\infty$-Lie algebra” which discusses the abstract concept and its models by dg-Lie algebras or by other things.
But anyway, let’s not fight over terminology anymore. Let’s save our energy for more substantial discussions!
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