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Can someone give me examples of mathematical objects which do not involve sets? For instance, the category of groups is a concrete category, but I want to consider non-concrete things.

Thanks

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    Are you just asking for a category which is not concretizable? A famous result of Freyd asserts that the homotopy category of topological spaces has this property. See, for example, http://amathew.wordpress.com/2012/01/26/homotopy-is-not-concrete/ .2012-12-17
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    I want something where no sets are involved or rather that sets are not a sufficient manner to model this mathematical object.2012-12-17
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    What do you mean by "involved"? As Peter Smith points out in his answer below, there are various things that you can talk about in a language that makes no explicit reference to set theory even if you can write down models of these things using sets. In what sense does a non-concretizable category not answer your question?2012-12-17
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    I have a decent intuition of sets. But I want some exposure to things which "are not sets". I want a mathematical object that cannot be realised using sets and thus has to be realised using "something else". The problem is that the "something else" is alien to me.2012-12-17
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    That sounds like you want examples of non-concretizable categories to me (but I'm biased; I think of all mathematical objects as living inside categories).2012-12-17
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    The example you gave of the homotopy category of topological spaces is a quotient of a concrete category. Can you give me an example of a non-concrete category that cannot be constructed (with taking a quotient an example of this) from a concrete category?2012-12-17
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    Properly defined it isn't a quotient – it's a localisation.2012-12-17

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