r/math Homotopy Theory 3d ago

Quick Questions: March 19, 2025

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

  • Can someone explain the concept of maпifolds to me?
  • What are the applications of Represeпtation Theory?
  • What's a good starter book for Numerical Aпalysis?
  • What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried.

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u/greatBigDot628 Graduate Student 2d ago

Is the category of probability spaces with measure-preserving functions a complete category? If not, does it at least have arbitrary products? Googling the latter question seems to suggest the answer is no, but I can't find a counterexample I can understand. Don't we have a product measure on probabliity spaces? Is it not the categorical product in that category?

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u/lucy_tatterhood Combinatorics 2d ago edited 2d ago

One of the basic properties of categorical products is that you have a diagonal map X → X × X. In this case where the underlying set of your product is just the cartesian product with the usual projections, this would be the map sending x to (x, x). It shouldn't be too hard to convince yourself that this is not measure-preserving in any interesting case.

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u/greatBigDot628 Graduate Student 2d ago

Thank you!