Ancestors

Toot

Written by Trebor on 2025-01-31 at 15:05

K(G,1) can be defined as the type of G-torsors, and addition corresponds to taking the tensor product (in the obvious way). So what's the multiplication when G is a ring?

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Descendants

Written by Trebor on 2025-01-31 at 15:14

wait... can it be...

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Written by Trebor on 2025-01-31 at 15:30

It's difficult to think about K(G,2) if it's defined as the truncation of suspension of K(G,1). Is there a better way?

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Written by daniel gratzer on 2025-01-31 at 17:33

@trebor So you can think of it as still classifying something like torsors, but now appropriately higher-dimensional. The relevant word here is "gerbe" (I think @buchholtz has a talk on such things in HoTT).

The fact that higher EM spaces classify n-gerbes is one of the results in higher topos theory (Theorem 7.2.2.26). It's also one of the motivations given in the preface, which is worth reading

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Written by daniel gratzer on 2025-01-31 at 17:39

@trebor @buchholtz On the other hand, the way that I usually think of it (for ordinary groups G, not sheaves of groups) is just that if G is abelian, it gives rise to an Eoo-space since it's discrete and therefore extremely coherent and so we can just deloop it as much as we want

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Written by Tom de Jong on 2025-01-31 at 17:41

@danielgratzer @trebor

Indeed @buchholtz does: https://youtu.be/eB6HwGLASJI

(See also Lemma 5.4 of our paper on acyclic types: https://arxiv.org/abs/2401.14106)

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Written by Naïm Camille Favier on 2025-01-31 at 15:32

@trebor I'm confused. Addition in G is loop concatenation in K(G, 1). What does that have to do with tensor products?

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Written by Trebor on 2025-01-31 at 15:34

@ncf There is a unique map K(G,1)^2 -> K(G,1) such that the action on the loop space is addition in G

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Written by Ulrik Buchholtz on 2025-01-31 at 20:57

@trebor check out this paper by @dwarn as well: https://arxiv.org/abs/2301.03685

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Written by David Wärn on 2025-02-01 at 11:19

@buchholtz @trebor In fact this question about representing the cup product in terms of torsors and gerbes was what originally led me to think about torsors and gerbes in HoTT. Eventually I became convinced that one should work with EM spaces representation-independently, just using their defining properties. So I became less interested in the original question. The paper linked above mentions the cup product but perhaps doesn't answer Trebor's question.

I think one answer would be as follows. Say we have groups A, B, C with C abelian, and a bilinear map μ : A → B → C. Say X is an A-torsor and Y is a B-torsor. Let's define the cup product of X and Y by its defining property, i.e. given a gerbe Z banded by C we say what it means for Z to be the cup product. It means we have a map h : X * Y → Z from the join of X and Y with the following property. Given points x₀ x₁ : X and y₀ y₁ : Y, we get a loop in X * Y that looks like glue(x₀,y₀) · glue(x₁,y₀)⁻¹ · glue(x₁,y₁) · glue(x₀,y₁)⁻¹. Now h sends this to a loop in Z. Since Z is banded by C, this loops corresponds to an element of C. The condition is that this element of C equals μ(x₁ - x₀, y₁ - y₀). But I don't have a proof that this works.

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