Natural transformations of A∞A_\infty-functors (between dg-categories) are “directed homotopies” (reference?)

Let A and B be dg-categories over a field, viewed as A∞-categories. The A∞-category (actually, dg-category) of strictly unital A∞-functors A→B will be denoted by Fun∞(A,B). It is described explicitly (in the non-unital case) for example in P. Seidel’s book (“Fukaya category and Picard-Lefschetz theory”). Let Δ1 be the 1-simplex category, namely, the linear category … Read more

Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety

Assume A is an Azumaya algebra of rank r2 on a smooth projective scheme Y over C. Let f:X→Y be the Brauer-Severi variety associated to A. I read here in a comment that the category of modules over A is equivalent to the categroy of modules on X which restrict to every fiber as a … Read more

reference request: category of crystals on a scheme is locally noetherian

I’m looking for a reference to the fact that the category of crystals on any scheme is locally noetherian. This is stated in Gaitsgory’s paper “Crystals and D-modules” but he doesn’t provide a reference. An idea of where to look would be very helpful, thanks very much. Answer AttributionSource : Link , Question Author : … Read more

Categories where every Mono Splits

When every epi splits a category is said to satisfy the Axiom of Choice. When every idempotent splits a category is called Cauchy Complete or Idempotent complete. These look to be well-studied notions, but what about the corresponding question for monos, i.e., the Axiom of Choice for the opposite category? Does it have a name … Read more

Galois categories and the connected components functor

In stacks 0BMQ, a Galois category is defined to be a functor F:C⟶FinSet such that C is finitely bicomplete, every object of C is a finite coproduct of connected objects, and F is exact and conservative. In a finitely extensive category C, finite coproduct decompositions are unique up to isomorphism if they exist, so if … Read more

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

A right adjoint to the truncated Witt functor?

For any ring A, let wEtA be the category of weakly etale A-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor Wr:wEtA⟶wEtWr(A) is well-defined and commutes with all colimits. Can one explicit a right adjoint to Wr ? Answer AttributionSource : Link , Question Author … Read more

Reflective factorization systems – from reflections to general adjunction?

In the context of Galois theory, the notions of “semi-left exact reflection” and “admissible adjunction” pop up. The notion of “simple reflection” is also crucial in simplifying theory by circumventing size conditions and simplifying factorizations. Section 4 of the paper Reflective subcategories, localizations, and factorization systems by Cassidy, Hébert, and Kelly characterizes simple reflections (Theorem … Read more

When are descriptions of formal unramifiedness/smoothness via lifting properties equivalent to those via induced arrows to pullbacks?

Formal unramifiedness of an arrow f:M→N in algebraic geometry or synthetic differential geometry in defined by asking the lifting problem below to have at most one solution (existence is not required). 1⟶M↓↗↓D⟶N In SDG, I think formal unramifiedness can also defined by asking the induced arrow TM→P to the pullback below to be a monomorphism. … Read more