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Added section In homotopy theory to nPOV.
This was written to go with this blog discussion. It's meant only as a first draft. Please have a look and improve!
I’ve slightly softened the commitment to the n-point of view on the basis that it’s really not a prerequisite for benefitting from or contributing to this Wiki, but more of a commonality between many of the largest contributors. I realize this is a foundational page so am trying to be light with the changes, but thought it better to just put something up rather than try to start what might be an unresolvable debate. Of course I won’t be offended if this is reverted.
kdcarlson
Around the nLab it is widely believed that higher algebra, homotopy theory, type theory, category theory and higher category theory provide a point of view on Mathematics, Physics and Philosophy which is a valuable unifying point of view for the understanding of the concepts involved.
How true is this statement with regards to today’s contributors to the nLab? Nathaniel Arkor on the category theory Zulip said that
The nPOV page was written in the early days of the nLab. A lot has changed since then in the attitudes to what is appropriate on the nLab.
There is a lot worth improving on in this old entry. But that paragraph you cite strikes me as plainly uncontroversial.
I’m not sure n-categories with n>1 help understanding eg harmonic analysis, or finite group theory, or enumerative combinatorics etc etc. There might be 1-categorical insights that can be brought to bear on these things, but going beyond that seems a strong claim. Elsewhere I’ve pointed out that n=1 is a legitimate value when talking about n-categories, but most people default I think to the generic case of n≥2 when they see the phrase.
You’ll have to admit that the example of groups, of all concepts, is a tautological case where homotopy theory provides a valuable unifying point of view.
The example of harmonic analysis also rather makes the opposite point – cf. categorical harmonic analysis.
More generally, a key insight of the last couple of decades was that a useful unifying point of view on anything 1-category theoretic is -category theory. It is in that vein that “type theory” can and does show up in the quoted paragraph even without the “homotopy”-adjective: It emerges.
In any case, I would invite anyone, who feels it is lacking, to add plain “category theory” to the list of concepts in that paragraph — if it were not already listed.
Harmonic analysis as in the Langlands program? Some higher category theory there.
The CFSG is indeed being reworked by Aschbacher to use 1-categories, at least. Not sure how -categories help with calculating the ATLAS character tables, myself… :-)
Group characters are a classical example where homotopy theory provides a valuable unifying point of view, cf. Hopkins, Kuhn & Ravenel 2000.
Following the example of the HomePage I replaced
higher algebra, homotopy theory, type theory, category theory and higher category theory
with
homotopy theory/algebraic topology, (homotopy) type theory, (higher) category theory and (higher) categorical algebra
as well as
which we may call the POV, the higher algebraic, homotopical, or n- categorical point of view.
with
which we may call the POV or the higher structures point of view, encompassing higher algebraic, homotopical, or n- categorical point of views.
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