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brushed up ringed topos a little, added the version over any Lawvere theory and linked it to “related concepts” (for use at Tannaka duality for geometric stacks).
touched also ringed space, locally ringed space, ringed site, structured (infinity,1)-topos and made sure they all link to each other
I have made the definition of morphisms of ringed toposes explicit at ringed topos.
I was going to type something on limits of ringed toposes, but maybe give me a sanity check:
the limit of a diagram of ringed toposes ( not locally ringed, mind you) should be formed by
forming the limit of the underlying toposes, ${\lim_\leftarrow}_i (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i}) \stackrel{p_i}{\to} (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$
then forming the colimit inside the limiting topos of the inverse images of all the internal ring objects ${\lim_\to} p_i^* \mathcal{O}_{\mathcal{X}_i} \in {\lim_\leftarrow}_i (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$.
Because with this the universal property is immediately checked. Unless I am mixed up.
I have tried to spell out details in a new subsection Properties – Limits and colimits.
That does seem to work. I wonder what the generality of that is.
That does seem to work.
Thanks for checking.
I wonder what the generality of that is.
I was wondering, too. But for the moment I am content with this simple case.
What I have found in the literature on this has been surprisingly disappointing. But I haven’t looked at Grothendieck’s original notes yet.
In addition to the related material in SGA4, let me mention that the relative schemes over ringed toposes were developed by M. Hakim in her thesis under Grothendieck’s guidance. That thesis also has a tiny bit of 2-categorical descent, I think.
By the way, interesting historical reminiscences from 2007 (Drinfel’d, Bloch, Illusie) about Grothendieck and his work with students, which I have not seen before: pdf.
I have added this to Alexander Grothendieck.
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