Constructs a modular ∞-operad of surfaces and a genus filtration showing 2D TFTs in a general target are modular algebras, producing a spectral sequence for mapping class group cohomology.
Arity Approximation of $\infty$-Operads
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abstract
Let $\mathbb{E}_d$ denote the little discs operad for $1 \le d \le \infty$ and let $\mathcal{C}$ be an $\infty$-category all of whose mapping spaces are $n$-truncated. We prove that when considering $\mathbb{E}_d$-monoids in $\mathcal{C}$, all coherence diagrams of arity $>n+3$ are redundant. More generally, for an $\infty$-operad $\mathcal{O}$ we bound the arity of the relevant coherence diagrams in terms of the connectivity of certain operadic partition complexes associated to $\mathcal{O}$.
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math.CT 1years
2025 1verdicts
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2-dimensional TFTs via modular $\infty$-operads
Constructs a modular ∞-operad of surfaces and a genus filtration showing 2D TFTs in a general target are modular algebras, producing a spectral sequence for mapping class group cohomology.