By Bertrand Toen, Gabriele Vezzosi

ISBN-10: 0821840991

ISBN-13: 9780821840993

This can be the second one a part of a sequence of papers known as "HAG", dedicated to constructing the principles of homotopical algebraic geometry. The authors begin via defining and learning generalizations of ordinary notions of linear algebra in an summary monoidal version class, resembling derivations, etale and gentle morphisms, flat and projective modules, and so forth. They then use their conception of stacks over version different types to outline a basic thought of geometric stack over a base symmetric monoidal version type $C$, and end up that this thought satisfies the predicted homes.

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**Extra resources for Homotopical Algebraic Geometry II: Geometric Stacks and Applications**

**Example text**

11 that an HA context is a triplet (C, C0 , A), consisting of a symmetric monoidal model category C, two full sub-categories stable by equivalences C0 ⊂ C A ⊂ Comm(C), such that: • 1 ∈ C0 , C0 is closed under by U-small homotopy colimits, and X ⊗L Y ∈ Ho(C0 ) if X and Y in Ho(C0 ). e. 6 are satisfied. Recall also that C1 is the full subcategory of C consisting of all objects equivalent to suspensions of objects in C0 , Comm(C)0 the full subcategory of Comm(C) consisting of commutative monoids whose underlying C-object is in C0 , and, for A ∈ Comm(C), A − M od0 (resp.

2. PRELIMINARIES ON LINEAR AND COMMUTATIVE ALGEBRA IN AN HA CONTEXT which has a natural structure of a graded π∗ (A)-module. This defines a functor π∗ : Ho(A − M od) −→ π∗ (A) − GM od, from Ho(A − M od) to the category of N-graded π∗ (A)-modules. 1. Let A ∈ Comm(C) be a commutative monoid in C, and a ∈ π0 (A). There exists an epimorphism A −→ A[a−1 ], such that for any commutative A-algebra C, the simplicial set M apA−Comm(C) (A[a−1 ], C) is non-empty (and thus contractible) if and only if the image of a in π0 (C) by the morphism π0 (A) → π0 (C) is an invertible element.

The second one is only valid for symmetric monoidal categories, and is a homotopy generalization of the notion of rigid objects in monoidal categories. 1. A morphism x −→ y in a proper model category M is finitely presented (we also say that y is finitely presented over x) if for any filtered diagram of objects under x, {zi }i∈I ∈ x/M , the natural morphism Hocolimi∈I M apx/M (y, zi ) −→ M apx/M (y, Hocolimi∈I zi ) is an isomorphism in Ho(SSet). 2. 1 has to be modified by replacing M apx/M with M apQx/M , where Qx is a cofibrant model for x.

### Homotopical Algebraic Geometry II: Geometric Stacks and Applications by Bertrand Toen, Gabriele Vezzosi

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