8
votes
3answers
466 views

References for the “nerve of an algebraic variety”

Let's do algebraic geometry over an arbitrary base ring $k$. I've frequently seen a definition of the algebraic $n$-simplex, as follows: $$\Delta^n = ...
4
votes
1answer
158 views

Cocycle condition for equivariant sheaves

Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$ : $X_n := G^n \times X$, $s_0:X_0 ...
2
votes
0answers
55 views

Relationship of height zero hypercovers to co-cartesian condition on cosimplicial modules

Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that ...
3
votes
2answers
219 views

When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$?

Let $A$ be a simplicial commutative ring over a field $k$ of characteristic zero (or a cdga in non-positive degrees with differential of degree -1). Let $M$ be a perfect $A$ module. If necessary, ...
21
votes
1answer
10k views

If I want to study Jacob Lurie's books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...
3
votes
0answers
149 views

What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem. Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...
0
votes
1answer
356 views

How does the discrete group act on simplicial set level by level

Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set? For example, if ...
4
votes
1answer
238 views

Local fibration vs. stalkwise fibration

Let $\mathbf{C}$ be a Grothendieck site with enough points. Let $p:\mathcal{E}\to \mathcal{F}$ be a map of simplicial presheaves on $\mathbf{C}$. Is it true that $p$ is a local (Kan) fibration if and ...