Is there a schema category for hyperstructures?

I am completely fascinated by Niels Baas’ notion of hyperstructures, chiefly because I can see how such gadgets could be used in modeling both biological and social systems, or other evolutionary systems (the basic intuition being that they describe complex systems obtained by “glueing” lower cells of arbitrary shapes). Now, here is my question: there … Read more

Nonabelian Poincare duality

Nonabelian Poincare duality is introduced by Jacob Lurie in “Higher Algebra”, Section 5.5.6. It seems, that my question is closely related to this definition. Question: what can one say about the functor X↦Σ∞Map(M,X) from spaces to spectra, if one knows, that M is a compact closed oriented manifold? By Greg Arone, “A generalization of Snaith-type … Read more

Free symmetric monoidal category of compactly generated category is compactly generated

Let k be a field and let C=StLink be the ∞-category of stable infinity categories enriched over the ∞-category Vectk, regarded as a symmetric monoidal ∞-category with unit object Vectk. The forgetful functor CAlg(C)→C admits a left adjoint Sym∗:C→CAlg(C). Now let C[z]=Sym∗(Vectk) – i.e. C[z] is the free stable symmetric monoidal category generated by Vectk. … Read more

Cohomology and higher structures

Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear G-action on M, there is an bijection between H2(G;M) and the set of extension E of G by M 1→M→E→G→1. Apparently, higher cohomologies correspond to longer extensions such as 1→M→F→E→G→1. Similar statements hold for Lie algebras. In Higher-Dimensional Algebra VI: Lie 2-Algebras, … Read more

Weakening the excision condition for spectra

Recently, I’ve noticed that the definitions of special Γ-spaces and spectra are quite close in spirit: Γ-spaces are pointed functors X:(Γop,⟨0⟩)→(S,∗) from Segal’s category to the category S of spaces. Moreover, we call X (see Definition 7.1 here) special if, for each ⟨n⟩,⟨m⟩∈Obj(Γ), the map X⟨n⟩∨⟨m⟩→X⟨n⟩×X⟨m⟩ induced by the inert surjections ⟨n⟩∨⟨m⟩→⟨n⟩ and ⟨n⟩∨⟨m⟩→⟨m⟩ is … Read more

Is there a 1-categorical treatment of operadic left Kan extensions in the literature?

Lurie develops in Section 3.1.2 of Higher Algebra a notion of operadic left Kan extension used to compute free algebras, giving a left adjoint $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\mathcal{O}’}(\mathcal{C})$ to the forgetful functor $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\mathcal{O}’}(\mathcal{C})$ associated to any map of $\infty$-operads $\mathcal{O}\to\mathcal{O}’$. There’s a number of constructions in the $1$-categorical setting that resemble this notion a bit, including monoidal Kan … Read more

Two Hattori-Stallings trace questions

Let R be a ring spectrum, and P a perfect R-module. The definition of THH(R) with a cyclic bar construction over the whole of Perf(R) allows to define a canonical morphism mapR(P,P)→THH(R), where mapR(P,P) is the mapping spectrum of P. Specializing to π0 yields a map [P,P]→π0THH(R). Let me call tr(f) the image of f:P→P … Read more

Quasi-categorical factorization system induced on XSX^S

Let X be an ∞-category (I am happy to assume it is bicomplete and stable, but this should not be necessary) and consider a factorization system F=(E,M) on X (this is defined in Section 24 of Joyal’s “Notes on quasi-categories“). In 24.10, Joyal mentions without proof that, given any simplicial set S, there is a, … Read more

Kan complexes and semigroups

Given a simplicial commutative semigroup: (1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group? (2) is the constant simplicial set on a set, Kan fibrant? A positive answer to (2) would give a negative answer to (1), since the constant … Read more

Interpolating between the flat and smooth affine lines in spectral algebraic geometry

Consider the following construction (which came up recently in a question about “spectral exterior algebras”): Pick a ring spectrum R and consider the ∞-category ModR of module spectra over R; Endow the ∞-category Fun(τ≤kS,ModR) with the Day convolution monoidal structure; Given an R-module M, take the constant functor ΔM:τ≤kS→ModR; Apply the free E∞-monoid functor to … Read more