Holmstrom Geometric category

See discussion in the concept note.

http://mathoverflow.net/questions/33477/did-durovs-work-give-an-example-of-noncommutative-schemes

http://mathoverflow.net/questions/58428/basic-questions-about-stacks

See Motivic stuff blog post.

There are categories which can be viewed as geometric objects. One example is Balmer’s tensor triangulated geometry, treated also in an arxiv preprint by Stevenson and Dell’Ambrogio. Another is Orlov-style geometry with derived categories of sheaves. Also: Tabuada’s noncommutative motives - is this based on DG categories?

http://ncatlab.org/nlab/show/Cartan+geometry

See Operad for a more on the PROs etc hierarchy.

Toen has a notion of geometric setting in the under Spec Z paper.

Toen notion of geometric context, see his cours in the Toen web folder, definition in cours2.pdf

Various cats of motives.

Fulton and MacPherson: Categorical framework for the study of singular spaces. In Geometry-Various folder. This seems to be the original source for bivariant theories in general, with quite a lot of material.

Generalized schemes; see Durov thesis link under Field with one element

http://ncatlab.org/nlab/show/Gabriel-Rosenberg+theorem

Teleman and Simpson: De Rham’s theorem for $\infty$-stacks.

http://mathoverflow.net/questions/56833/riemannian-manifolds-etc-as-locally-ringed-spaces

http://golem.ph.utexas.edu/category/2010/06/vladimir_arnold_12_june_1937_3.html mentions Arnold’s ideas on various notions of geometry

Question: Can everything be subsumed by simplicial sheaves on a site?

http://ncatlab.org/nlab/show/geometry+(for+structured+(infinity,1)-toposes) (interesting)

arXiv:1103.2139 Localization of ringed spaces from arXiv Front: math.AG by W. D. Gillam Let $X$ be a ringed space together with the data $M$ of a set $M_x$ of prime ideals of $\O_{X,x}$ for each point $x \in X$. We introduce the localization of $(X,M)$, which is a locally ringed space $Y$ and a map of ringed spaces $Y \to X$ enjoying a universal property similar to the localization of a ring at a prime ideal. We use this to prove that the category of locally ringed spaces has all inverse limits, to compare them to the inverse limit in ringed spaces, and to construct a very general $\Spec$ functor. We conclude with a discussion of relative schemes.

http://mathoverflow.net/questions/84641/theme-of-isbell-duality

Demazure: Lectures on p-divisible groups. LNM302. Abstract intro to schemes, group schemes, formal schemes and more. Perhaps the reference I was really looking for is Demazure and Gabriel: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs.

Topological concrete category at nlab, with discussion of some terminology

nice topological space nlab

http://ncatlab.org/nlab/show/stratifold

http://nlab.mathforge.org/nlab/show/orbifold

http://ncatlab.org/nlab/show/scheme

http://nlab.mathforge.org/nlab/show/formal+scheme

arXiv:0907.3925 Compactly Generated Stacks: A Cartesian-Closed Theory of Topological Stacks from arXiv Front: math.AG by David Carchedi A convenient 2-category of topological stacks is constructed which is both complete and Cartesian closed. This 2-category, called the 2-category of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of 2-categories between compactly generated stacks and those classical topological stacks which admit locally compact atlases. Compactly generated stacks are also equivalent to a bicategory of topological groupoids and principal bundles, just as in the classical case. If a classical topological stack and a compactly generated stack have a presentation by the same topological groupoid, then they restrict to the same stack over locally compact Hausdorff spaces and are homotopy equivalent.

arXiv:1101.2796 A-schemes and Zariski-Riemann spaces from arXiv Front: math.AG by Satoshi Takagi In this paper, we will investigate further properties of A-schemes. The category of A-schemes possesses many properties of the category of coherent schemes, and in addition, it is co-complete and complete. There is the universal compactification, namely, the Zariski-Riemann space in the category of A-schemes. We compare it with the conventional Zariski-Riemann space, and characterize the latter by a left adjoint.

nLab page on Geometric category