Linked Questions

18 votes
17 answers
6k views

Vector spaces without natural bases

Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural". To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...
29 votes
6 answers
3k views

Why are finiteness conditions important (and how to recognize them)?

I think everybody here has met lots of finiteness conditions, like those requiring a vector space to be finite dimensional, an abelian group to be finitely generated, a ring to be Noetherian, a ...
11 votes
3 answers
896 views

$\infty$-ary tensor product on a category

Is there any notion in the literature which captures the idea of an $\infty$-ary tensor product on a category $C$? This should include tensor products of $\alpha$-indexed families of objects in $C$ ...
Martin Brandenburg's user avatar
15 votes
5 answers
1k views

Monoids with infinite products

Say a monoid $M$ has infinite products if, for any (possibly infinite) sequence $(m_1,m_2,\ldots)$ of elements of $M$, there exists an element $m_1m_2\cdots\in M$, satisfying some good properties. ...
David Spivak's user avatar
  • 8,549
14 votes
1 answer
1k views

Infinite tensor product of states

Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher. Even in the simplest ...
Glacier's user avatar
  • 143
1 vote
3 answers
809 views

Limit for divergent sequences

Sorry for the title, but I think it's funny. Can you write down a homomorphism (of additive groups) $\mathbb{R}^\mathbb{N} \to \mathbb{R}$, which is nontrivial and whose kernel contains the finite ...
Martin Brandenburg's user avatar
5 votes
1 answer
295 views

(Z/n)^(I) is a direct summand of (Z/n)^I

Dear group theorists, Let $n \geq 1$ and $I$ be an infinite set (you may assume $I$ to be countable). Is the abelian group $(\mathbb{Z}/n)^{(I)}$ (direct sum of copies $\mathbb{Z}/n$) a direct ...
Martin Brandenburg's user avatar
3 votes
1 answer
806 views

cardinality of product modulo direct sum

let $(X_i)_{i \in I}$ be an infinite family of sets with $|X_i| \geq 2$. we define an equivalence relation on $X = \prod_{i \in I} X_i$ by $x \sim y \Leftrightarrow \{i : x_i \neq y_i\}$ is finite. ...
Martin Brandenburg's user avatar
6 votes
1 answer
613 views

Generalization of analytic functors

It's been a long time since I posted the following question on stackexchange. Now I think it's better to adress it to you, in the hope I will reach the right audience: Martin's comment, albeit useful,...
fosco's user avatar
  • 13k
6 votes
1 answer
551 views

Split powers of the multiplicative group of a field

Let $K$ be a field, $K^\times$ its multiplicative group and $I$ an infinite set. Is then $(K^\times)^{(I)} \subseteq (K^\times)^I$ a direct summand? If not, is it possible to characterize the fields ...
Martin Brandenburg's user avatar
2 votes
2 answers
404 views

Is certain topology-related set a distributive lattice?

In my research the following problem appeared (and if it is true, this solves positively several my conjectures): Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ($n$ ...
porton's user avatar
  • 739