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Categories of abstract smooth models and their singular envelopes. (English) Zbl 0831.18005

From the introduction: “The building-up of objects with singularities from the corresponding smooth models and their maps is a process which occurs, in some specific form, in each one of the usual geometries: algebraic \((\ldots)\), analytic \((\ldots)\), differential \((\ldots)\), mixed \((\ldots)\). A prototype is, for instance, the formation of the category of complex spaces from the category of open sets in \(\mathbb{C}^n\), \(n \geq 0\), and of holomorphic maps. This paper deals with the axiomatization of this process”.
A category over topological spaces is a category \({\mathcal S}\) together with a functor \(\pi : {\mathcal S} \to {\mathcal T} {\mathcal O} {\mathcal P}\). A morphism from such a category \(({\mathcal S}, \pi)\) to another \(({\mathcal S}', \pi')\) is a functor \(\eta : {\mathcal S} \to {\mathcal S}'\) such that \(\pi = \pi' \circ \eta\). Let \({\mathcal S}\) be a category over topological spaces. If \(D\) is an object of \({\mathcal S}, D_0\) will stand for the topological space \(\pi (D)\) and \([D]\) for its underlying set. Similarly, if \(\alpha\) is a morphism in \({\mathcal S}, \alpha_0\) will stand for the continuous map \(\pi (\alpha)\) which is also denoted by \([\alpha]\) when continuity is forgotten. Accordingly, the functor obtained by the composition of the forgetful functor of \({\mathcal T} {\mathcal O} {\mathcal P}\) with \(\pi\) is denoted by \([\cdot]\).
Now the author defines the category of (abstract) smooth models: A small category \({\mathcal S}\), over topological spaces, is said to be a category of (abstract) smooth models if:
\((s_1)\) \({\mathcal S}\) has an initial object \(I\) with \(I_0 = \emptyset\) but \(D_0 \neq \emptyset\) when \(D \neq I\);
\((s_2)\) any object \(D\) in \({\mathcal S}\) has sufficiently many open subjects;
\((s_3)\) for each pair of objects \(D,E\) in \({\mathcal S}\), \(h_D (E)\) is a sheaf.
A smooth-model category \({\mathcal S}\) is said to be additive if:
\((s_4)\) \({\mathcal S}\) has finite products and a terminal object \(e\) with \(e_0=\{0\}\) and such that the map
\(\text{hom} (e,D) \ni \varepsilon \to \varepsilon_0 (0) \in [D]\) is bijective for all \(D\).
If \({\mathcal S}\) is a smooth-model category, then an extension of \({\mathcal S}\) is a pair \(({\mathcal S}', \eta)\) consisting of a category \({\mathcal S}'\) over topological spaces and a morphism \(\eta : {\mathcal S} \to {\mathcal S}'\) such that:
1. \({\mathcal S}'\) satisfies conditions \((s_1) - (s_3)\) and has a glueing property, and 2. \(\eta\) is fully faithful and preserves open subobjects.
Let \({\mathcal S}\) be an additive smooth-model category. An extension \(({\mathcal S}',\eta)\) of \({\mathcal S}\) is called a singular envelope of \({\mathcal S}\) if:
\((\sigma_1)\) \(\eta\) preserves finite products;
\((\sigma_2)\) every parallel pair in \({\mathcal S}'\) which proceeds from \({\mathcal S}\) (via the functor \(\eta\)) has an equalizer which is \({\mathcal S}\)- stable.
\((\sigma_3)\) every object of \({\mathcal S}'\) is locally isomorphic to the equalizer of parallel pairs in \({\mathcal S}'\) proceeding from \({\mathcal S}\).
The first main result is a relative version of the Yoneda representation theorem which the author states as follows: Let \({\mathcal S}\) be a category of smooth models (not necessarily additive). An \({\mathcal S}\)-functored space \(X\) is semilocal iff its Yoneda morphism \(\rho_X\) is an isomorphism.
The second main result is an existence and uniqueness theorem for the singular envelope: Any additive smooth-model category \({\mathcal S}\) has a singular envelope which is unique up to equivalence. Moreover, if \((\widehat {\mathcal S}, \eta)\) is a singular envelope of \({\mathcal S}\), then:
1. \(\widehat {\mathcal S}\) has finite limits and the canonical functors \(\widehat {\mathcal S} \to {\mathcal S}\)-loc and \([\cdot] : \widehat {\mathcal S} \to {\mathcal S} {\mathcal E} {\mathcal T}\) preserve them;
2. \(\widehat {\mathcal S}\) satisfies condition \((s_4)\);
3. If the structure functor \(\pi : {\mathcal S} \to {\mathcal T} {\mathcal O} {\mathcal P}\) preserves finite products, then the structure functor \(\widehat {\mathcal S} \to {\mathcal T} {\mathcal O} {\mathcal P}\) preserves all finite limits.
In fact the author proves the existence of a canonical process which associates with each additive smooth-model category \({\mathcal S}\) a singular complex \({\mathcal S}\)-an of \({\mathcal S}\), whose objects are called \({\mathcal S}\)- analytic spaces. The author remarks that most of the fundamental categories of geometry are of the form \({\mathcal S}\)-an (up to equivalence). As an example the author gives the Grothendieck complex spaces and he also introduces here two other such categories: Banach differentiable spaces and Banach mixed spaces.
The reviewer considers that this paper is a fundamental work in the contemporary mathematics, its form is very clear and the working not sophisticated.
Reviewer: I.Pop (Iaşi)

MSC:

18F99 Categories in geometry and topology
32C15 Complex spaces
58B99 Infinite-dimensional manifolds
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References:

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