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Is there a category theory condition such that we can say when a property of a category also holds for it's sub-category?

For example, in the catgeory of groups $f$ being bijective implies it is an isomorphism, and in the sub-categories of rings, modules, algebras and so forth. However this is not the case for example for topological vector spaces.

My question is there a categorical condition (aside from simply defining a new term), that can ensure a property will hold for a sub-category?

Keen-ameteur
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  • First of all, there is no such thing in categories as a "bijection." A bijection is something that makes sense when talking about the category of sets. So we can say that a category $\mathcal C$ has a notion of bijection if there is a functor $F:\mathcal C\to \mathcal{Set},$ and then a map $m$ in $\mathcal C$ is an $F$-bijection of $F(m)$ is an isomorphism in $\mathcal{Set}.$ In the cases you've mentioned, there are "standard" functors $F,$ but there can be more than one such $F.$ – Thomas Andrews Apr 16 '19 at 15:19
  • Well they are sub-categories of $Set$, in which I can say what is a bijection by the appropriate forgetful functor. – Keen-ameteur Apr 16 '19 at 15:26
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    Technically, $Group$ is not a sub-category of $Set$ because the objects in $Group$ are not sets, but sets with a binary function. Two different groups can have the same set. You can work around this with a function $F:Group\to Set$ which sends a group $(G,\times)$ to a more specific set that is not $G$, but that amounts to choosing such a Functor $F.$ – Thomas Andrews Apr 16 '19 at 15:39
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    @Keen-ameteur: The categories of groups, rings, modules, etc., are not subcategories of $\mathbf{Set}$. For example, a set can have more than one group structure. It looks like you're interested in properties of concrete categories that are reflected by their forgetful functors. For example, to say that a group homomorphism is an isomorphism if and only if it is a bijection, is exactly to say that the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$ reflects isomorphisms. – Clive Newstead Apr 16 '19 at 15:39
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    For example, in the category of topological spaces, the forgetful functor $(X,\tau)\to X$ does not preserve bijections, but one might be able to find another functor $F:\mathbf{Top} \to \mathbf{Set}$ that sends homeomorphisms to bijections. – Thomas Andrews Apr 16 '19 at 15:46
  • So is $Rng$ not a sub-category of $Grp$ as well (also technically)? In general, does adding a structure makes such a structure not a sub-category in the technical sense? – Keen-ameteur Apr 16 '19 at 17:49
  • @ThomasAndrews Is that what you meant to say? I'm pretty sure a bijection on spaces induces a bijection on underlying sets :) – Kevin Carlson Apr 17 '19 at 22:20

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Yes, one example is that a full subcategory of an abelian category is again abelian (under certain additional conditions):

Full subcategory of abelian category is abelian

Dietrich Burde
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  • So is the category of topological vector spaces not an abelian one? – Keen-ameteur Apr 16 '19 at 15:28
  • Exactly, this category is additive, but not abelian, see this question, i.e., the comment by Matt E. – Dietrich Burde Apr 16 '19 at 15:31
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    Note that in the linked question there are additional conditions to ensure that the full subcategory is abelian, and these are necessary : for example, the full subcategory of $\mathbf{Ab}$ containing just the object $\mathbb{Z}$ is not abelian, and even a subcategory that is additive and has kernels and cokernels need not be abelian (see for example this question). – Arnaud D. Apr 16 '19 at 16:56