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I'm beginning learning category theory and seeing some examples I wonder whether (or why) is necessary that arrows preserve some structure of objects. I know that there are functors that "forget" some information of a category but my point is to do this inside the category. For example can I define a category $Top^*$ with the class of all topological spaces as objects and functions as arrows, or a category $Rng^*$ with rings as objects and group homomorphisms as arrows?. I know I doesn't make much sense to study this ('cause what's the point of objects in $Top^*$ have topologies), however my the question is if category definitions (or axioms) allow this classes.

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    $\begingroup$ As long as your category satisfies the axioms, you should not have any problems. However, the naming convention will be disturbing and you would not gain anything by doing this. $\endgroup$ – Hans Feb 17 '17 at 20:29
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    $\begingroup$ Yes those are totally valid categories; you can check easily that they satisfy the axioms of a category. People just don't study them because they're not interesting $\endgroup$ – Alex Mathers Feb 17 '17 at 20:30
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    $\begingroup$ What your arrows are changes a few things. Category theory is full of universal objects, like limits, products, initial and terminal objects and so on. If you have the same category, but change the arrows, then the universal objects change. For a specific example, in the category of unital, commutative rings with ring homomorphisms that preserve $1$, coproducts are tensor products. If you add in all group homomorphisms to the category, coproducts become direct sums instead. $\endgroup$ – Arthur Feb 17 '17 at 20:54
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    $\begingroup$ Actually, it's quite common to be more liberal about the morphisms in a category than we are about the objects. E.g., in the category of metric spaces we allow arbitrary continuous maps as morphisms: the metric-preserving maps (isometries) still give a category, but a much less useful one. Polish spaces are another interesting example where we forget some structure on the objects (namely a metric of a certain sort). $\endgroup$ – Rob Arthan Feb 17 '17 at 21:07
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The definition of category does not require that objects have any kind of structure or that arrows preserve structure. You can define a category by drawing some dots and putting some arrows between dots to satisfy the requirements for composition, existence of identities, and associativity.

The definition of category is abstract in the sense that it says a category has objects and arrows and imposes axioms on the objects and arrows.

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