442:
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are the set-theoretic limits with the final and initial sigma-algebra respectively. Canonical examples of direct and inverse systems are the ones arising from
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of a pair of morphisms is given by placing the induced sigma-algebra on the subset given by the set-theoretic equalizer. Dually, the
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The split epimorphisms are (up to isomorphism) the measurable surjective maps of a measurable space onto one of its retracts.
208:). Moreover, since any function between discrete or between indiscrete spaces is measurable, both of these functors give
74:, or other notations. Some authors also restrict the category only to particular well-behaved measurable spaces, such as
334:
352:
427:
450: – Definition and properties of the category of Markov kernels, in more detail than at "Markov kernel".
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which equips a given set with the indiscrete or trivial sigma-algebra. Both of these functors are, in fact,
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699:
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The split monomorphisms are (essentially) the inclusions of measurable retracts into their ambient space.
323:
605:"A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics"
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which assigns to each measurable space the underlying set and to each measurable map the underlying
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uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in
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Category of measurable spaces, on the model of the page "Category of topological spaces".
654:
LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in
Computer Science
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of two measurable maps is again measurable, and the identity function is measurable.
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are given by placing particular sigma-algebras on the corresponding (co)limits in
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is given by placing the quotient sigma-algebra on the set-theoretic coequalizer.
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541:. Lecture Notes in Mathematics. Vol. 915. Springer. pp. 68–85.
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which equips a given set with the discrete sigma-algebra, and a
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650:"Probability monads with submonads of deterministic states"
439: – Category in mathematics where the objects are sets
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Journal of
Logical and Algebraic Methods in Programming
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Pages displaying wikidata descriptions as a fallback
568:"From probability monads to commutative effectuses"
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535:"A categorical approach to probability theory"
106:preserving this structure. There is a natural
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393:are the isomorphisms of measurable spaces.
279:(considered as a measurable space) is the
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58:N.B. Some authors reserve the name
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353:filtrations in probability theory
244:. In fact, the forgetful functor
62:for categories whose objects are
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428:Category of topological spaces
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690:Categories in category theory
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584:10.1016/j.jlamp.2016.11.006
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448:Category of Markov kernels
443:Category of measure spaces
409:(and therefore also not a
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631:10.1016/j.aim.2020.107239
516:Moss & Perrone (2022)
413:) since it does not have
389:measurable maps, and the
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672:10.1145/3531130.3533355
609:Advances in Mathematics
234:complete and cocomplete
603:Fritz, Tobias (2020).
533:Giry, Michèle (1982).
291:measurable space is a
138:The forgetful functor
82:As a concrete category
566:Jacobs, Bart (2018).
381:measurable maps, the
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316:product sigma-algebra
76:standard Borel spaces
295:. There are thus no
460:Measurable function
415:exponential objects
238:limits and colimits
224:Limits and colimits
700:Probability theory
547:10.1007/BFb0092872
556:978-3-540-11211-2
320:Cartesian product
200:are equal to the
108:forgetful functor
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192:(meaning that
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578:: 200–237.
518:, p. 3
480:Giry (1982)
339:coequalizer
142:has both a
53:composition
21:mathematics
684:Categories
663:2204.07003
622:1908.07021
526:References
387:surjective
43:and whose
639:201103837
467:Citations
379:injective
335:equalizer
324:coproduct
289:singleton
277:empty set
271:include:
240:exist in
104:functions
45:morphisms
31:, is the
422:See also
385:are the
377:are the
248: :
232:is both
173: :
152: :
133:function
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405:is not
318:on the
308:product
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322:. The
287:; any
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658:arXiv
635:S2CID
617:arXiv
411:topos
216:into
90:is a
551:ISBN
403:Meas
375:Meas
369:The
347:and
333:The
312:Meas
306:The
301:Meas
285:Meas
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269:Meas
258:Meas
250:Meas
242:Meas
230:Meas
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196:and
179:Meas
158:Meas
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88:Meas
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373:in
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188:to
175:Set
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246:U
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150:D
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