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The category of fields, denoted Field, is the full subcategory of CRing whose objects are fields.

In Field, the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the prime field is an initial object.
[...] A curious aspect of the category of fields is that every morphism is a monomorphism

In Field we have no initial or terminal objects but this is not true if we are in the subcategory of fields of fixed characteristic because we can say that a prime field is an initial object.

I want to understand better this situation: before we have neither initial/terminal object nor zero object but then we can 'extract' an initial object if we move into a "subcategory of fields of fixed characteristic". What makes this possible?

So.. what structures/morphism we have in $\text{field}\rightarrow\text{subcategory of fields of fixed characteristic}$ to return a initial object as prime field ?
I don't understand well the purpose of this 'subcategorification'

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  • $\begingroup$ I’m assuming you’re in the category where maps must preserve multiplicative identities? $\endgroup$ – Randall Dec 28 '18 at 17:33
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If morphisms preserve the identity of the field, then there are no morphisms between fields of different characteristics. Hence the category of fields is presented as disjoint union (coproduct) of its subcategories $Field_0, Field_1, \ldots$, where $Field_p$ stands for the subcategory of fields of characteristic $p$. Such a "subcategorification" nothing more than taking direct summand. There is an initial object in each summand, while there is no "global" initial object, because the summands are "disjoint".

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  • $\begingroup$ I think "connected component" is more appropriate than "direct summand"; in fact these subcategories are the largest connected subcategories of the category of fields. $\endgroup$ – Arnaud D. Dec 29 '18 at 14:21
  • $\begingroup$ Each category is a coproduct of its connected components, which are "coproduct-irreducible", hence one can speak about "summands" as well. $\endgroup$ – Ilya Vlasov Dec 29 '18 at 14:38

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