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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.
A weak indexing system is almost essentially unital (or aE-unital) if for all non-contractible , the empty -set is in .
A weak indexing system is unital if for all .
Let be the subgroup lattice of . We say that a subposet is a 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 furnishes an equivalence of posets
It is not clear how to generalize the subgroup lattice to orbital -categories. Instead, one should note that closure under conjugation implies that a transfer system only depends on its image under the homogeneous G-set functor . Thus we may make the following definition.
A core-containing wide subcategory is a \emph{transfer system} if, for all diagrams in with in and whose map is a summand inclusion, the map is in .
Note that closure under summands implies that, whenever is a unital weak indexing category, its intersection is a transfer system. Nardin-Shah 22 generalized the correspondence between indexing systems and transfer systems to orbital -categories, and Stewart 24 generalized this to weak indexing systems as the following.
The monotone map has a fully faithful right adjoint whose essential image is spanned by the -indexing systems.
The poset of subcommutative G-∞ operads containing corresponds with ; these are called N-∞ operads (see there for details).
(…)
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, Combinatorial operads [arXiv:1705.03585v3]
On those coming from equivariant linear isometries operads:
Jonathan Rubin, Characterizations of equivariant Steiner and linear isometries operads [arXiv:1903.08723v2]
Usman Hafeez, Peter Marcus, Kyle Orbsby, Angélica Osorno, Saturated and linear isometric transfer systems for cyclic groups of order , [arXiv:2109.08210]
Ethan MacBrough, Equivariant linear isometries operads over Abelian groups, [arXiv:2311.08797]
Julie E. M. Bannwart, Realization of saturated transfer systems on cyclic groups of order by linear isometries -operads [arXiv:2311.01608]
On enumerating categories of transfer systems
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]
Evan E. Franchere, Kyle Ormsby, Angélica Osorno, Weihang Qin, Riley Waugh, Self-duality of the lattice of transfer systems via weak factorization systems [arXiv:2102.04415]
Scott Balchin, Ethan MacBrough, Kyle Ormsby, Lifting operads from conjugacy data, [arXiv:2209.06798]
Scott Balchin, Ethan MacBrough, Kyle Ormsby, The combinatorics of operads for and , [arXiv:2209.06992]
Over orbital categories:
Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra [arxiv:2203.00072]
Natalie Stewart, Orbital categories and weak indexing systems [arXiv:2409.01377]
Last revised on September 5, 2024 at 19:00:43. See the history of this page for a list of all contributions to it.