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location  Münster, Germany  
age  27  
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I am interested in the interactions between algebraic geometry and category theory. More specifically, I "model" algebraic geometry on cocomplete symmetric monoidal categories.
Here is a link to my PhD thesis. Comments are welcome.
Email: [my last name] [at] unimuenster.de
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Notation: Categories of measur(abl)e spaces
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Notation: Categories of measur(abl)e spaces
Actually $\mathsf{Bor}$ looks really good. But then it's still open how to denote the category of measure spaces. (Or the category of measure spaces equipped with a $\sigma$ideal, as suggested by Dmitri Pavlov.) 
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Is there an introduction to probability theory from a structuralist/categorical perspective?
One can talk about a field without having to write it down as a tuple. Just say $K$ is the field, an object of the category of fields and not of the category of sets, and don't confuse it with its underlying set $K$. Similarly, we can talk about graphs without having to think of them as pairs of sets. The answer claims that it is somehow difficult or distracting to define graphs that way, but I don't agree with this either. 
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Is there an introduction to probability theory from a structuralist/categorical perspective?
By the way, I disagree with Rudin's comment. It basically says "It is a tradition to ignore forgetful functors, so I will follow this tradition" and "argues" for this procedure by saying that otherwise one would "have to" imagine the reals as a quadruple. No. It is really about the question in which category one works. There is a field of real numbers, there is a measure space of real numbers, there is a topological space of real numbers, etc., and it is very unfortunate that all are denoted by the same symbol. At least, one should not forget the forgetful functors between these categories. 
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Is there an introduction to probability theory from a structuralist/categorical perspective?
"probability theory is not about probability spaces"  yes, but it is perhaps about the category of probability spaces and measurable maps. Random variables are morphisms in that category. The approach to look at families of random variables to understand some distribution is really in the spirit of the Yoneda Lemma. Morphisms are more important than objects. 
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Notation: Categories of measur(abl)e spaces
Thank you! I didn't know this alternative terminology. "measurable space" seems to be more modern than "Borel space". Is this correct? 
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Notation: Categories of measur(abl)e spaces
@JohannesHahn: Yes, I would also call it the (general) 'transformation formula'. 
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Notation: Categories of measur(abl)e spaces
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Notation: Categories of measur(abl)e spaces
@DmitriPavlov: Thank you for your comment. I have found several papers on measure theory and probability theory which apply category theory to the categories of measurable resp. probability spaces. Do you say that it is hard to do nontrivial measure theory with these categories, but that it is possible for other categories, or that category theory is not useful for nontrivial measure theory at all? In the former case, I would like to know which alternative categories are more suitable. 
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Notation: Categories of measur(abl)e spaces
What do you think about $\mathsf{Measure}$ for the category of measure spaces (and $\mathsf{Meas}$ for the category of measurable spaces)? 
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Notation: Categories of measur(abl)e spaces
@MichaelGreinecker: Thanks! The authors even say "This gives a category which is often called $\mathsf{Meas}$.". So it should be common. Have you seen a notation for the other category? 
Mar 24 
asked  Notation: Categories of measur(abl)e spaces 
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Most striking applications of category theory?
I agree that this combination of methods is really amazing and a good advertisement for all these abstract concepts. 
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awarded  Nice Question 
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How is the monoidal product defined on the functor category between symmetric monoidal dagger cats
Pointwise, obviously? But the symmetric monoidal structure on $C$ is not used, in fact it's irrelevant. 
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Brandt's definition of groupoids (1926)
Yes I read the category mailing list, and this is also a partial motivation for my question. The second paragraph says that there was an influence, the third paragraph says that there was none. Hm. 
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Definition of internal field objects
$T \to X$ is a morphism in $\mathcal{C}$, obviously. $R$ is a ring object, not $X$. It is important that $T$ is an arbitrary object, not just $1$. 