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When we say the category of groups, do we refer to the same category of groups? For example, in Tom's mind, there is a category of groups; in Jack's mind, there is a category of groups. However, one day Tom constructs a group $A$ nobody else knows, thought $A$ is isomorphic to some group $B$ contained in Jack's category of groups, can we regard $A$ contained in Jack's category of groups? When we say a set of people, we mean the same set of people, though we don't know everybody who lives on the earth, but the existence of everybody is objective. But the existence of group $A$ is not objective, it only exists in Tom's mind. Then how do we correctly understand the category of groups?

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    $\begingroup$ Categories (considered as being encoded objects in some fixed metatheory, say $\mathsf{NBG}$) are actual objects, and shouldn't change from person to person. In the case of groups, one could argue that maybe your definition of a group does not agree with another persons definition of a group (e.g. perhaps your signature for groups is $(\cdot,-^{-1},e)$, while another's only has $\cdot$). That case, what you call $\mathbf{Grp}$ and what they call $\mathbf{Grp}$ would technically be different, but they're isomorphic as categories. $\endgroup$ – Hayden Sep 13 '18 at 4:53
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    $\begingroup$ "Category" has a technical meaning, and the category of groups (with group homomorphisms as maps) has a very specific definition that uniquely identifies it. $\endgroup$ – Malice Vidrine Sep 13 '18 at 4:53
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    $\begingroup$ ...That said, ideally everyone is really referring to the 'same' category of groups. The idea that a group 'only exists in Tom's mind' is really not (I think) a question we ask while doing mathematics, being much more philosophical in nature. I think the idea that we really have some fixed metatheory in the background that we implicitly (or perhaps explicitly) use should sweep aside those questions from a mathematical perspective. $\endgroup$ – Hayden Sep 13 '18 at 4:56
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    $\begingroup$ Categories and groups are just smoke and mirrors here. How can we talk about the set of all natural numbers? It does not exist objectively either, and there are surely numbers never seen or thought of by anybody. $\endgroup$ – Ivan Neretin Sep 13 '18 at 6:07

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