/sci/ - Science & Math

>>12457035
The nerve of a group represents singular cohomology in the sense that [math][X,NG] \cong H^1(X;G)[/math]. Is there a similar theorem for when [math]G[/math] is a groupoid (e.g. the fundamental groupoid of some other space)?

>>11593150
It is well-known that you can describe groups, rings, modules, etc. internal to any category using so-called finite product theories. This is generalized to finite limit theories, which also include internal categories, internal groupoids, etc. Any coalgebraic structure can be obtained in a straightforward way by simply dualizing.

What I want to know is if any work has been done in the "non-Cartesian" setting; i.e. replacing categorical products with an arbitrary monoidal product.

Currently learning about Hopf algebroids on the way to the E-Adams spectral sequence, for context.

>>10803629
Are any anons here good with simplicial sets?

I've heard the claim that the Hurewicz map comes from the prolongation [math]F:\mathbb{sSet} \rightarrow \mathbb{sAb} [/math] of the free abelian group functor. I'm having trouble understanding what is meant by this. The Hurewicz map should be a natural transformation [math]\pi_n(-) \Rightarrow H_n(-)[/math]. I know that for a topological space [math]X[/math], the singular simplicial set [math]Sing(X)[/math] has the same homotopy groups calculated in [math]sSet[/math], and that the homotopy groups of [math]F(Sing(X))[/math] calculated in [math]sAb[/math] are just the homology groups of [math]X[/math]. My guess would be that since [math]F[/math] is part of an adjunction [math](F,U)[/math] for [math]U[/math] the forgetful functor, the Hurewicz map should come from the unit [math]\text{Id} \Rightarrow UF[/math] of this adjunction. I can't see how to make this work though, since applying [math]U[/math] to [math]F(Sing(X))[/math] takes you out of [math]sAb[/math]. Thoughts?

>>9916453
Describe the irreducible components of the following algebraic sets in [math]\mathbb{C}^3[/math]

[eqn]\mathcal{V}(y^2-xz,x^4-yz,z^2-x^3y)[/eqn]

[eqn]\mathcal{V}(xz-y^2,z^3-x^5)[/eqn]

I know to look at the primary ideal decomposition of the ideals involved here, but I have zero intuition when it comes to breaking them down. Any tips/general strategies for problems like this?