A categorical question - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T17:59:30Z http://mathoverflow.net/feeds/question/11425 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11425/a-categorical-question A categorical question Mehdi Omidali 2010-01-11T12:39:11Z 2010-01-11T15:38:28Z <p>Hi everyone, I want to know the descriptive translation and explanation of a paragraph of EGAI, (chapter 0, 3.1.3)</p> <p>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\$.</p> <p>Thanks</p> http://mathoverflow.net/questions/11425/a-categorical-question/11439#11439 Answer by Zavosh for A categorical question Zavosh 2010-01-11T15:38:28Z 2010-01-11T15:38:28Z <p><em>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\$</em>.</p> <p>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 <em>Chapter IV: Structures</em> of <em>Theory of Sets</em>.</p> <p>Here first Grothendieck defines a <em>solution to a universal problem</em> 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.</p>