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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.
David 6, Urs 7
Of course, as time progresses, new connections are made and higher categories will be ever more useful even for more classical concepts.
However, the structure of Lab is in some details making too much heavy weight on 1-categorical reader by sometimes putting to yet not attested hypothetical and baby -context. (One of the reasons why I had hard time to convince even my graduate students to regularly use it.) Most notably, many topics in very technical algebra of very specific 1-categorical kind have table for “context” algebra but in fact over 90% in table is higher algebra. For example, the entries Wedderburn-Artin theorem, Jacobson radical or even the entry radical.
I would suggest (and I can materialize the first version if approved) that we have a separate context algebra where more entries of relevance for the usual 1-categorical algebra were included and then separately the context higher algebra. Then 1-algebra items can have 1 (one) link to higher algebra (unless the higher concept is already in substantial use) and this is enough if somebody wants to switch to higher context. For example, Jacobson radical should have the table to 1-algebra, but term “ring” should have also to higher algebra as 2-rings are already much in use/interest and it is about a basic and not specialized notion, so the general context is more relevant. Otherwise it is just hard to navigate for somebody not familiar to higher categories or, more often, interested to find quickly relevant links/details in the usual algebra. For example, it is natural to list Wedderburn-Artin theorem as one of the cornerstones of classical algebra in the 1-algebra table, rather than monoidal Dold-Kan correspondence which is however more important in higher algebra.
All the context-menus leave room for much improvement and are waiting for people to work on them.
I see that somebody else started overhaul of the algebra contents in the meantime. I will join to improve soon, today working on an issue in localization theory.
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