I am looking for more examples of universal constructions in probability theory. Like the construction a of Gaussian space from a real Hilbert space, or a Poisson jump process from a measurable space with a $\sigma$-finite measure. There must be tons of examples, even though their universality (in the sense of category theory) is probably not commonly emphasized.
6
-
$\begingroup$ Related: mathoverflow.net/questions/20740 and mathoverflow.net/questions/117274 $\endgroup$– Martin BrandenburgFeb 5, 2013 at 19:06
-
$\begingroup$ @Martin: Thanks! $\endgroup$– UwFFeb 6, 2013 at 8:43
-
1$\begingroup$ I was thinking of examples that could convince classical probabilists that it might be worthwile for them to study some basic category theory... but I might be dreaming ;) $\endgroup$– UwFFeb 6, 2013 at 15:39
-
1$\begingroup$ @UwF Now that the dust has settled, what is your take on your own "dream"? That is, what would be some "examples that could convince classical probabilists that it might be worthwile for them to study some basic category theory"? $\endgroup$– DidSep 28, 2013 at 9:01
-
1$\begingroup$ It is clearly not an answer, but it is related : a categorical approach to measure theory. $\endgroup$– PeceDec 7, 2013 at 14:40
|
Show 1 more comment
1 Answer
$\begingroup$
$\endgroup$
3
The category of measurable spaces is topological this means that they have initial & final structures analogously to those in topology. These can be universally expressed as noted on the wikipedia page.
Cylinder set measures are defined categorically if not universally, and are used to define measures on infinite-dimensional spaces such as the abstract wiener space construction.
Lebesgue measure and the integral can be defined universally as shown by Tom Leinster.
-
$\begingroup$ The notes by Leinster are cool, which book of Barr and Wells does he refer to? How technical does the construction of the initial object get, if one fills in all the details? $\endgroup$– UwFFeb 12, 2013 at 9:03
-
$\begingroup$ The notes by Leinster cannot be found. Please correct the url. $\endgroup$ Dec 3, 2013 at 7:54
-
$\begingroup$ I think this is the right link maths.ed.ac.uk/~tl/pssl/leinster.pdf $\endgroup$– UwFDec 3, 2013 at 17:43