• The category of (co)cartesian structures on a \(G\)-category with finite indexed (co)products is contractible.
• Cartesian \(\mathcal{O}\)-algebras can be presented as "\(\mathcal{O}\)-monoids;" in particular, \(\mathcal{O}\)-algebras in the catesian structure on coefficient systems are Segal objects over either of the associated algebraic patterns.
• If \(\mathcal{C}^{\otimes}\) is cocartesian over all arities at which a reduced \(G\)-operad \(\mathcal{O}^{\otimes}\) has nonempty structure spaces and the underlying \(G\)-category of colors of \(\mathcal{O}\) is contractible, then objects of \(\mathcal{C}\) admit contractible moduli of \(\mathcal{O}\)-algbera structures; moreover, the converse is true, if you generalize to unital \(I\)-operads.
• If \(I\) is almost essentially unital and \(\mathcal{O}^{\otimes}\) is almosst essentially reduced, then there is an equivalence \(\mathcal{O} \otimes \mathcal{N}_{I\infty} \simeq \mathcal{N}_{I\infty}\) if and only if \(A\mathcal{O} \leq I\). Moreover, the assumptions were necessary: if \(I\) is not almost essentially unital, then \(\mathcal{N}_{I\infty}^{\otimes 2}\) is not connected, let alone equivalent to \(\mathcal{N}_{I \infty}^{\otimes}\). Hence almost-unital weak \(\mathcal{N}_\infty\)-operads are idempotent algebras, i.e. they classify smashing localizations.
• There is an equivalence \(\mathcal{O} \otimes \mathcal{N}_{I\infty} \simeq \mathcal{O}\) if and only if the underlying \(I\)-operad of \(\mathcal{O}\) is cocartesian--equivalently, if and only if \(\mathcal{O}\)-algebra \(G\)-spaces have \(I\)-indexed Wirthmüller isomorphisms.
• The above point allows you to compute \(\mathcal{N}_{I\infty} \otimes \mathcal{N}_{J \infty} \simeq \mathcal{N}_{I \vee J \infty}\) in the almost-unital setting, affirming the remaining conjecture of Blumberg-Hill.
• \(\otimes\)-idempotence of \(\mathrm{Comm}^{\otimes}_G\) allows for the sliced equifibered perspective to be symmetric monoidal, leading to an easy construction of a canonical lift of the Boardman-Vogt tensor product to a presentably symmetric monoidal \(G\)-\(\infty\)-category of \(G\)-operads.
• Lurie's Disintegration and assembly procedure works for \(G\)-space colored \(G\)-operads, on the level of giving natural \(G\)-colimit presentations via one-color \(G\)-operads; \(G\)-functorial distributivity of \(\otimes\) allows us to compute tensor products of \(G\)-space colored \(G\)-operads in terms of one-colored \(G\)-operads. Put a pin in this--we'll use it for equivariant Dunn additivity with tangential structure.
• \(A \mapsto \mathrm{RMod}_A\) takes \(\mathcal{O} \otimes \mathbb{E}_1\)-algebras to \(\mathcal{O}\)-monoidal \(\infty\)-categories; in particular, right modules over an \(I\)-commutative algebra are given a natural \(I\)-symmetric monoidal structure when \(I\) is an indexing category.
• Factorization homology is \(G\)-symmetric monoidal; in particular, the above computation lifts \(\mathrm{THR}\) to a natural endofunctor of \(\mathbb{E}_{\infty\sigma}\)-rings. As a bit of service, we show how to lift this to a lax \(C_2\)-symmetric monoidal cyclotomic structure, constructing a lax \(C_2\)-symmetric monoidal functor whose induced endofunctor of \(C_2\)-commutative rings is Quigley-Shah's Real topological cyclic homology.

• The underlying \(G\)-symmetric sequence is monadic.
• There is a well-behaved localizing subcategory of \(G\)-\(d\)-operads (whose structure spaces are \((d-1)\)-truncated), compatible with all of the constructions.
• The \(\mathcal{O}\)-\(G\)-coefficient systems functor \(\mathcal{O} \mapsto \mathrm{Alg}_{\mathcal{O}}(\mathrm{Coeff}^G\mathcal{C})\) detects equivalences on the base change of the underlying \(G\)-symmetric sequence to \(\mathcal{C}\); in particular, evaluation on \(n\)-truncated \(G\)-spaces detects \(n\)-equivalences of \(G\)-operads and evaluation on \(G\)-spaces is conservative.
• The equivariant operadic nerve intertwines everything in sight, so it has a conservative right-derived functor; moreover, it's only the coherences that make things annoying, so it's easy to verify that it's an equivalence on one-color \(G\)-1-operads.
• The Boardman-Vogt tensor product works as you expect it to and intertwines with the Day convolution structure on \(G\)-symmetric monoidal \(\infty\)-categories via the envelope.

This paper is an exposé on weak indexing systems, the combinatorics behind weak \(\mathcal{N}_\infty\) operads. The main point here is to show that, even though it's somewhat more complicated, you can run some version almost all of the homotopical combinatorics you know and love in the more general setting of subterminal \(G\)-operads, perhaps under mild unitality assumptions.

Submitted to tAMS in May, 2025.

In ongoing work with Branko Juran, we hope to use equivariant higher algebra to fulfill the conjectures of Blumberg-Hill-Mandell in the \(\infty\)-categorical setting, replacing strictly commutative algebras with normed algebras (having finite-index norms).

In ongoing work with Gregoire Marc, we hope to show that the Nardin-Shah and Bonventre-Pereira models for \(G\)-operads agree, ultimately verifying that one can use "monoids in \(G\)-symmetric sequences" in equivariant higher algebra arguments exactly when it's convenient.

In a looping Mainz project, Bikramjit Kundu, Alex Waugh, and I have started to work out the rational cohomology of equivariant iterated loop spaces, as well as the integral cohomology operad of \(\mathbb{E}_V\), in hopes of approaching an equivariant Cohen's theorem.

This is now ongoing work with Christian Carrick, Mike Hill, Ryan Quinn, and Qi Zhu.

After doing so, I hope to add \(\mathbb{R}\) and \(C_2\), as well as replace \(3\) with \(2 \rho - 1\), in section 2 of Hahn-Wilson and claim the new result as my own.

Lower bounds on the volumes of hyperbolic link complements are given via a new construction: a bracelet link is a link in \(S^3\) decomposed as a cycle of interconnected tangles, and it is proved that a bracelet link of \(2n\) tangles, such that each individual tangle may be replicated into a hyperbolic bracelet link of \(2n\) copies of the tangle, is hyperbolic, with volume at least the average of the replicated links.https://atwater.caltech.edu/team-member/miles-johnson

This replication is generalized to arbitrary 3-manifolds via a construction called starbursts, which separate the manifold into pieces, which have a well defined \(2n\)-replicant; if the resulting pieces from removing a regular neighborhood of a starburst have hyperbolic replicants, then the 3-manifold is hyperbolic, with volume at least the average of the volumes of the replicants.

Applications are presented to hyperbolicity of links in thickened surfaces and in the solid torus.