Linked Questions
31 questions linked to/from Is there an introduction to probability theory from a structuralist/categorical perspective?
333
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Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
49
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28
answers
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Problems where we can't make a canonical choice, solved by looking at all choices at once
It's a common theme in mathematics that, if there's no canonical choice (of basis, for example), then we shouldn't make a choice at all. This helps us focus on the heart of the matter without giving ...
53
votes
4
answers
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When is $L^2(X)$ separable?
I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in ...
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68
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2
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Is there a category structure one can place on measure spaces so that category-theoretic products exist?
The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...
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37
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5
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Reference for the Gelfand duality theorem for commutative von Neumann algebras
The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann ...
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27
votes
3
answers
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Why is free probability a generalization of probability theory?
Note: This question was already asked on Math.SE nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope ...
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17
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4
answers
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Good introduction to statistics from a algebraic point of view?
There are already lots of questions on this subject like
Is there an introduction to probability theory from a structuralist/categorical perspective?
Is there a combinatorial/topological treatment ...
- 283
26
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2
answers
3k
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Corollaries of the Yoneda Lemma in Analysis?
This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: https://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis.
I am looking for some ...
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25
votes
1
answer
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Is the opposite category of commutative von Neumann algebras a topos?
By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict ...
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15
votes
2
answers
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Analytical origins of the Stone duality
I've asked this question in the HSM community, but by the nature of my question, some user told me to ask this question here.
This is the original post https://hsm.stackexchange.com/q/13087/14296
...
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11
votes
2
answers
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Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
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3
votes
3
answers
654
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Free probability with unbounded random variables?
This is partially inspired by this question and this blog post.
When trying to express classical probability in the "free probability" setting one takes an algebra of random variables equipped with ...
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10
votes
2
answers
2k
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random category theory
This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references ...
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10
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2
answers
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Why Kleisli Markov categories and not the Eilenberg-Moore categories of the associated monads
Why is there so much interest in the Markov categories which are Kleisli categories for monads corresponding to distributions etc. but not much discussion of the E.M. categories?
For example, the E.M. ...
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12
votes
1
answer
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Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
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