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there was no references here; have added now pointer to
I think another popular example would be passing from the category of chain complexes to the derived category. The only reference I can think of off the top of my head is May and Ponto, where they talk about different model structures on chain complexes.
The uses of derived categories in algebraic geometry (leaning toward representation theory) almost never talk about how and why you can localise a category. So I can’t think of any references relating to that particular use.
This entry is about “homotopy localization” in the sense of $\mathbb{A}^1$-localization.
(Maybe it’s not the best entry title. I think what happened is that orgininally there was an entry “$\mathbb{A}^1$-localization” which eventually became “motivic homotopy theory” and then I needed another entry to be able to point specifically to the general concept of localization at an interval object.)
The general concept that you seem to be thinking of is in localization and localization of an (infinity,1)-category etc.
Ah ok that makes much more sense. The title is a bit confusing :>)
Okay, I have added a disambiguation line at the beginning of the entry:
This entry is about the specific notion of localization “at an object” (at an interval object) as in (but not exclusive to) motivic homotopy theory. For the general concept see instead at localization or localization of an (infinity,1)-category and related entries.
Seems to me like the page should be renamed. “localization at at interval object”?
Strictly speaking, the full term would be “localization at left-homotopy equivalences of a given (interval) object”, where I put “(interval)” in parenthesis, because for the definition of the maps being inverted, we don’t even need it to be equipped with an interval structure.
Maybe one could try “$I$-homotopy localization”, which would reduce to ” “$\mathbb{A}^1$-homotopy localization” for $I = \mathbb{A}^1$.
I don’t really like incorporating a particular notation, like $I$ for the object, into the name of a page.
From a HoTT perspective I would call this “internal nullification”. External nullification at an object $I$ is localizing at the map $I\to \ast$, so that the local objects are the ones for which $Map(\ast,Z) \to Map(I,Z)$ are equivalences, i.e. every map $I\to Z$ is uniquely constant. Internal nullification expresses a similar statement in the internal type theory, hence the local objects are those for which the analogous map of internal hom-objects $Z \cong Z^{\ast} \to Z^I$ is an equivalence. Using the internal-hom adjunction this is equivalent to being local in the usual sense for $X\times I \to X$ for all objects $X$.
Internal nullifications are the prime examples of (idempotent monadic) modalities. Somehow I’m not sure I really noticed before that the basic objects of motivic homotopy theory are just the modal types for a modality on an $(\infty,1)$-topos.
So it’s a variant of shape modality? The second example at homotopy localization should mention that, no?
I second Mike’s suggestion of localisation at an interval object; even though not strictly correct in terms of describing which arrows are inverted, the effect in motivic homotopy theory is to make the interval contractible, so I think it sort of acquires the correct kind of meaning.
Yes, it is indeed a very close variant of a shape modality! Just as in real-cohesive HoTT shape is nullification at the real numbers, in “motivic HoTT” shape is nullification at $\mathbb{A}^1$. That’s really intriguing: for real-number cohesion we start by knowing the notion of “discrete homotopy type” and construct shape so as to reflect back into it, whereas in the motivic world we have to “invent” the corresponding notion of “discrete motivic homotopy type” as a corresponding nullification.
If you don’t like “$I$-homotopy localization” I can offer “geometric homotopy localalization”.
The word “nullification” seems a little inappropriate to me, a) since it alludes to abelian/stable structure (null, zero) and b) since we are not inverting just a single morphism $I \to \ast$, but all the projections $X \times I \to X$ out of domains for left $I$-homotopies.
But I don’t care too much about the entry title, feel free to rename the entry if you feel the need. Please just make sure to keep the current name as redirect.
I don’t like “homotopy localization” (with or without prefix) because all localization (of this sort) is homotopy-theoretic; it’s not clear that the word is meant to refer specifically to “geometric homotopies”. What’s different here is that (1) we are inverting an object (in the form of a map $I \to \ast$) rather than a morphism and (2) we are inverting it “internally” in the sense that the map on internal hom-objects $Z \to Z^I$ is required to be an equivalence rather than just the external hom-spaces $Map(\ast,Z) \to Map(I,Z)$. The word “nullification” has already been generalized to the unstable context, e.g. it appears in Hircshhorn’s book; and the “internal” perspective means that we don’t need to refer to projections $X\times I \to X$ for arbitrary $X$, that’s only necessary if we insist on talking about it as an external localization.
Maybe something incorporating the word “shape”? Or perhaps “localization at geometric homotopies”?
Or perhaps “localization at geometric homotopies”?
That’s okay with me. I am surprised a little that you suggest that while rejecting the permutation “geometric homotopy localization”, but I am fine with it either way.
The internal perspective is neat, but since that’s not what’s traditionally understood it seems a little too esoteric for the general name of the concept.
My problem with “geometric homotopy localization” is that I naturally parse it as “geometric (homotopy localization)”. Saying “localization at geometric homotopies” makes the associativity clear.
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