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

Homotopy invariance of the moduli stack of smooth $G$-bundles?

This question ought to have a straightforward (perhaps even glaringly obvious) answer, but so far I’ve already wasted a few hours trying to untangle this web of inconsistent identifications. I’m sure someone more experienced in this area will be able to quickly point out exactly where things go wrong. Let $G$ be a Lie group … Read more

Terminology for filtered ∞\infty-categories

Often to prove that a simplicial set X∙ is a contractible ∞-groupoid, we instead prove that X∙ is a contractible Kan complex (i.e. satisfies the extension property for inclusions ∂Δn↪Δn), which is clearly a stronger property. I find myself in the situation of proving that a simplicial set X∙ is a filtered ∞-category by proving … Read more

Coherence laws when composing 2-monads

To have the composition of two monads be a monad itself, we need a distributive law natural transformation satisfying certain coherence laws. I’m interested in the strict 2-monad case, i.e. a strict 2-functor equipped with unit and counit natural transformations that satisfy the zig-zag equations on the nose. I presume in such a case it’s … Read more

Are there obstructions to refining partially coherent natural transformations up-to-homotopy between quasicategories?

EDIT: It looks like I didn’t define the simplicial set R∼Δ1 clearly below. It’s not simply Ho(R)Δ1. Rather, an n-simplex of R∼Δ1 is a 1-simplex of Ho(RΔn), explicitly, a pair of n-simplices X0,X1 together with a homotopy class [α] of an edge between them. So R∼Δ1 is a partially coherent simplicial set of arrows of … Read more

Extending morphisms in an $A_{\infty}$ category to natural transformations

Suppose we are given a small (enriched) category $C$, and for $a,b \in C$ an isomorphism $m:a \to b$. It is always possible to find a functor $F: C \to C$, with $F(a)=b$ and a natural transformation $N$ of the identity functor to $F$ so that $N_a=m$. This is a nice simple exercise. Can this … Read more

The lisse-etale site and derived algebraic geometry

If one reads say Olsson’s book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff’s of the cotangent complex). Now it could well be that I did not read close enough, but my impression is that in Toen-Vezzosi’s Homotopical Algebraic Geometry II and … Read more

Can the cobordism hypothesis be formulated as a statement about adjoint functors?

I would like to formulate the cobordism hypothesis for general tangential structure as a statement about adjoint (∞,1)-functors. For a space Y with an action of O(n) let X=Y×O(n)EOn and let ζ be the pullback of the universal bundle under the map X→BOn. Then I will write Bord Yn for what is usually called Bord(X,ζ)n. I … Read more

Is there a good, general description of morphisms right orthogonal to effective epimorphisms?

Let C be a locally presentable, locally cartesian closed ∞-category. Then I think it’s not hard to show that the class of effective epimorphisms in C is closed under colimits and cobase-change, and is accessible, so that it forms the left class of a factorization system. If C is an ∞-topos, then the corresponding right … 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