3 ms·

What is category in this sense?

earthicus 8y ago
a baire category is an old-timey synonym for a baire space. It's related to topology & analysis, not algebra (like category theory). In the modern theory we can make an analogy (but it is just an analogy!) between 'categories' of (complete non-empty) metric spaces on the one hand, and measure spaces (of positive measure) on the other, as follows: metric spaces: measure spaces: first category zero measure second category positive measure residual full measure baire measurable
- mlevental 8y ago
  i remember in undergrad my analysis prof saying there were two types of analysis: hard analysis (sans baire category theorem) and soft analysis (using baire category theorem). i wonder if what he actually meant was in baire spaces and not in baire spaces.
  - earthicus 8y ago
    I suspect your professor was referring to 'qualitative' vs 'quantitative' approaches when referring to soft vs hard analysis. The baire category theorem is intimately related to these to approaches, and leads to three theorems (1) the uniform boundedness principle, (2) the open mapping theorem, and (3) the closed graph theorem, that establish equivalences between the qualitative and quantitative theories. According to Tao's textbook, in practice these are not used much directly, instead one starts with quantitative bounds, then derives the qualitative corollaries. The baire category theorem tells us why those there theorems are true, and thus tells us why we can take that step from quantitative to qualitative. I think this is what your professor meant.
    - mlevental 8y ago
      can you give an example of a qualitative corollary? do you mean a characterization of a space or something like that?
      - earthicus 8y ago
        Referring to properties of linear operators. Examples are finiteness, surjectivity, non-negativity, monotonicity, min/max principles, contraction properties, etc.
- Sniffnoy 8y ago
  A terminology note, I think "first category"/"second category"/"residual" have largely been replaced by the clearer "meagre"/"non-meagre"/"comeagre". Or at least, I hope they've largely been so replaced, because geez the old terminology is a pain. :P
  - mcguire 8y ago
    Clearer?!?
    - antidesitter 8y ago
      I'd say so. "non-meagre" = not meagre = positive, and "comeager" = complement of meagre = full. The only thing you have to remember is what "meagre" means.
    - Sniffnoy 8y ago
      Yes. "Meagre" means small. Which is what a first-category set is, but the name "first category" doesn't tell you that (how is one supposed to remember which of first and second category is which?). And then after that, you don't need to introduce any more words. If you mean to say something is not meagre, you just say it's not meagre, same as you would with any concept. If you mean to say it's complement is meagre, you just say it's comeagre, same as with anything. The names "first category" and "second category" tell you nothing, not even making it obvious which is which -- and moreover, they make it sound like the two are both equally useful sides of a distinction, like finite and infinite, while in reality one of these is actually directly useful and the other is just everything that isn't that (which is how most distinctions in mathematics go; you wouldn't make up a separate new word for "not compact", for instance). And then "residual" makes it sound like there's some new thing going on, when of course it's really just being co-first-category. But co-first-category's quite the mouthful, isn't it...?

a baire category is an old-timey synonym for a baire space. It's related to topology & analysis, not algebra (like category theory). In the modern theory we can make an analogy (but it is just an analogy!) between 'categories' of (complete non-empty) metric spaces on the one hand, and measure spaces (of positive measure) on the other, as follows: metric spaces: measure spaces: first category zero measure second category positive measure residual full measure baire measurable

mlevental 8y ago
i remember in undergrad my analysis prof saying there were two types of analysis: hard analysis (sans baire category theorem) and soft analysis (using baire category theorem). i wonder if what he actually meant was in baire spaces and not in baire spaces.
- earthicus 8y ago
  I suspect your professor was referring to 'qualitative' vs 'quantitative' approaches when referring to soft vs hard analysis. The baire category theorem is intimately related to these to approaches, and leads to three theorems (1) the uniform boundedness principle, (2) the open mapping theorem, and (3) the closed graph theorem, that establish equivalences between the qualitative and quantitative theories. According to Tao's textbook, in practice these are not used much directly, instead one starts with quantitative bounds, then derives the qualitative corollaries. The baire category theorem tells us why those there theorems are true, and thus tells us why we can take that step from quantitative to qualitative. I think this is what your professor meant.
  - mlevental 8y ago
    can you give an example of a qualitative corollary? do you mean a characterization of a space or something like that?
    - earthicus 8y ago
      Referring to properties of linear operators. Examples are finiteness, surjectivity, non-negativity, monotonicity, min/max principles, contraction properties, etc.
Sniffnoy 8y ago
A terminology note, I think "first category"/"second category"/"residual" have largely been replaced by the clearer "meagre"/"non-meagre"/"comeagre". Or at least, I hope they've largely been so replaced, because geez the old terminology is a pain. :P
- mcguire 8y ago
  Clearer?!?
  - antidesitter 8y ago
    I'd say so. "non-meagre" = not meagre = positive, and "comeager" = complement of meagre = full. The only thing you have to remember is what "meagre" means.
  - Sniffnoy 8y ago
    Yes. "Meagre" means small. Which is what a first-category set is, but the name "first category" doesn't tell you that (how is one supposed to remember which of first and second category is which?). And then after that, you don't need to introduce any more words. If you mean to say something is not meagre, you just say it's not meagre, same as you would with any concept. If you mean to say it's complement is meagre, you just say it's comeagre, same as with anything. The names "first category" and "second category" tell you nothing, not even making it obvious which is which -- and moreover, they make it sound like the two are both equally useful sides of a distinction, like finite and infinite, while in reality one of these is actually directly useful and the other is just everything that isn't that (which is how most distinctions in mathematics go; you wouldn't make up a separate new word for "not compact", for instance). And then "residual" makes it sound like there's some new thing going on, when of course it's really just being co-first-category. But co-first-category's quite the mouthful, isn't it...?