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Math Help - Question on Category Theory

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    Question on Category Theory

    As a preliminary note, I have absolutely no background in category theory. The concept was very briefly introduced in an introductory topology book I was reading as motivation for the introduction of groups, and it was left with very little description. Anyway, my question is the following - when speaking of the category of topological spaces, the book described the objects as topological spaces (as expected), but it stated that the morphisms were continuous maps. Here's where I'm confused. Is it simply convenient in further studies to define the morphisms as continuous mappings, or is it necessary to do so?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Math Major View Post
    As a preliminary note, I have absolutely no background in category theory. The concept was very briefly introduced in an introductory topology book I was reading as motivation for the introduction of groups, and it was left with very little description. Anyway, my question is the following - when speaking of the category of topological spaces, the book described the objects as topological spaces (as expected), but it stated that the morphisms were continuous maps. Here's where I'm confused. Is it simply convenient in further studies to define the morphisms as continuous mappings, or is it necessary to do so?
    This is very vague. What do you mean "further studies"? The general rule of thumb is that the morphisms in a category tend to be the structure preserving maps (duh), so why does it surprise you that the morphisms for topological spaces are continuous maps? For other categories though the morphisms are not going to be continuous maps. For \textbf{Grp} they'll be homomorphisms, etc.
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    My question is much more basic than that. What I'm asking is, did the author introduce the morphisms of Top as being continuous maps because they have to be continuous, or did he introduce them as being continuous maps because when working with the morphisms of Top, the continuous maps are the important ones to study?

    I guess more fundamentally, what I'm asking is would allowing the morphisms to just be any map between topological spaces be an invalid choice?
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    Nevermind, I see, morphisms are supposed to be structure preserving. That was the little caveat I was unclear on - the only rules for morphisms I was given is that they had to be associative and there had to be an identity.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Math Major View Post
    Nevermind, I see, morphisms are supposed to be structure preserving. That was the little caveat I was unclear on - the only rules for morphisms I was given is that they had to be associative and there had to be an identity.
    I don't know entirely much about category theory, but this isn't a caveat per se. It's analogous to asking why not, given the set of equivalence classes \mathbb{Z}_6, define a bijection between it and S_3 and define the group structure to be that which makes the bijection an isomorphism? Because it's not what's natural or useful. That is my limited impression.
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    Alright, that's all I was wondering, if the choice of morphisms was made because it's useful or there was some mandatory requirement that I was missing.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Math Major View Post
    Alright, that's all I was wondering, if the choice of morphisms was made because it's useful or there was some mandatory requirement that I was missing.
    It's quite arbitrary. Continous mappings are chosen in this case because they're the kind of mapping one usually studies between topological spaces; of course we can have maps between them as sets, but then there's no point in thinking of them as topological spaces.

    There are some pretty wild categories. You can create pretty much any category you want, as long as the axioms are respected. Whether it will be interesting or not is another question!
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