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symmetric monoidal (∞,1)-category of spectra
An indexing system is a combinatorial datum which uniquely determines an N-∞ operad.
In the following definition, fix an orbital ∞-category and its finite-coproduct closure.
For instance, may be the orbit category of a finite group, in which case is the category of finite G-sets.
A full -subcategory is called a weak indexing system if
(objects) whenever the -value is nonempty, it contains the -set .
(composition) for all , is closed under -indexed coproducts.
is called an indexing system if, additionally,
(coproducts) for all and ,
A subcategory is called a weak indexing category if it satisfies the following conditions:
(restriction-stability) morphisms in is stable under pullbacks along arbitrary maps in .
(Segal condition) A pair of maps and are in if and only if their coproduct is in ; and
(-action) contains all automorphisms of its objects.
A weak indexing category is called an indexing category if it contains the fold map for all and .
Given a weak indexing category, we may define a full -subcategory
The assignment furnishes an equivalence between the posets of weak indexing categories and weak indexing systems; this restricts to an equivalence between indexing categories and indexing systems.
Let be the subgroup lattice of . We say that a subposet is a \emph{transfer system} if it is closed under conjugation and restriction.
Given an indexing system, we let denote the subposet consisting of inclusions such that the corresponding map is in . The following theorem was independently proved by Rubin 2017 and by Balchin, Barnes & Roitzheim 2019.
is a transfer system, and this outlines an equivalence of posets
The poset of subcommutative G-∞ operads containing ] corresponds with ; these are called *[[N-∞ operads* (see there for details).
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Original discussion:
Further characterization:
Andrew Blumberg, Michael Hill, Incomplete Tambara functors, Algebr. Geom. Topol. 18 (2018) 723-766 [arXiv:1603.03292, doi:10.2140/agt.2018.18.723]
Jonathan Rubin, Characterizations of equivariant Steiner and linear isometries operads [arXiv:1903.08723v2]
Scott Balchin, David Barnes, Constanze Roitzheim, -operads and associahedra, Pacific J. Math. 315 (2021) 285-304 [arXiv:1905.03797, doi:10.2140/pjm.2021.315.285]
Over orbital categories:
Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra [arxiv:2203.00072]
Natalie Stewart, Orbital categories and weak indexing systems (draft) [pdf]
Last revised on June 24, 2024 at 20:27:40. See the history of this page for a list of all contributions to it.