# A categorical question [closed]

Hi everyone, I want to know the descriptive translation and explanation of a paragraph of EGAI, (chapter 0, 3.1.3)

Supposons que $K$ soit la catégorie définie par une « espèce de structure avec morphismes » $\Sigma$, les objets de $K$ étant donc les ensembles munis de structures d'espèce $Sigma$ et les morphismes ceux de $S$.

Thanks

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## closed as off topic by Qiaochu Yuan, Reid Barton, Loop Space, Charles Siegel, José Figueroa-O'FarrillJan 11 '10 at 15:29

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Sigma=$\Sigma$ S at the end of the paragraph is again $\Sigma$ –  Mehdi Omidali Jan 11 '10 at 12:41
There are many, many online translation services that could do this for you. –  Qiaochu Yuan Jan 11 '10 at 12:42
Yes, but translation services are usually quite bad at technical language. Here's a translation; althugh it's really so close to the English that it's a bit embarrassing to write it down: "Let us suppose that K is the category defined by a "kind of structure with morphisms" Sigma, the objects of K being thus the sets provided with a structure of the kind Sigma and the morphisms those of Sigma. –  José Figueroa-O'Farrill Jan 11 '10 at 15:28
Google Translate learns if you correct it, so it will get better at technical language as more mathematicians use it. –  Ben Webster Jan 11 '10 at 18:40

## 1 Answer

We suppose that $K$ is a category defined by a "species of structures with morphisms" $\Sigma$, the objects of $K$ are thus the sets endowed with structure of species $\Sigma$ and the morphisms those of $\Sigma$.

This is the language of Bourbaki. "Species of structures" are Bourbaki's version of categories. The analogue of an object is for example a set with some "structure", like a topology, or a group operation. Morphisms are functions that "preserve the structure". Rigorous definitions of these things and their morphisms, in full Bourbaki generality, can be found in Chapter IV: Structures of Theory of Sets.

Here first Grothendieck defines a solution to a universal problem in a category $K$. He's actually defining a limit, but for diagrams that look like a cover. He also defines a sheaf with values in an arbitrary category $K$, giving the extra condition on a presheaf. Then he restricts to the case where $K$ is made of objects with extra structures as above, and assumes that $K$ satisfies a further property: that a solution to a universal problem in $K$ remains a solution in the category of sets if we forget the structures. Then he says that in this case a sheaf with values in $K$ is also a sheaf of sets.

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Thanks, I couldn't understand if it mean "a category of objects with a certain structure" or "a category of objects that their structures can be extracted from morphisms", and your response is completely clear for me. –  Mehdi Omidali Jan 11 '10 at 17:32