• 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 is not yet submitted: click here for v1.5.

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.

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.

I hope to construct canonical lifts of $\mathbb{E}_{2n}-\mathrm{MU}$-algebra structures on $\mathrm{BP}$ to $\mathbb{E}_{n \rho}-\mathrm{MU}_{\mathbb{R}}$-algebra structures on $\mathrm{BP}_{\mathbb{R}}$ using the cellularity results announced in Hill-Hopkins as well as the above conjectures concerning change of group functors applied to equivaraint operadic cotangent complexes.

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.

Rubin constructed an associative version of $N_\infty$ operads; these are the family of $\mathbb{E}_1$-containing subfunctors of an equivariant associative operad. Tensor products of these are more complicated, as they are not tensor closed (indeed, $\mathbb{E_2}$ is not a subfunctor of the equivariant associative operad, as it is not discrete)-- I hope to say something about this using techniques reminescent of Schlank-Yanovski and Guillou-May; in particular, I conjecture that the $k$-fold tensor product of $\mathrm{As}(I)$ can be interpreted as having grouplike algebras which are the $k$-fold deloopings for the maps in the transfer system associated to $I$, in the sense of limits of $S$-indexed cubes, and when $k = \infty$, this computes $\mathcal{N}_{I \infty}.$

The aim of this is to supplant the $\mathbb{E}_V$ family with a family whose tensor-indecomposables are of a combinatorial nature, bringing the explicit computational ease of the case $G=C_2$ to arbitrary groups. This is low priority, since my computational examples of interest currently only have $C_2$ actions.

What remains in the equivariant version of the program to identify all models of operads with each other is an equivariant lift of Chu-Haugseng-Heuts; I hope to use the general machinery of algebraic patterns to add a $G$ everywhere in the above paper.