Abstract
We define the categories of (abstract) smooth models (Definition 1.2) and, in the additive case, their singular envelopes (Definition 1.5). The first main result is a relative version of the Yoneda representation theorem (Theorem 1.6), and the second one is an existence and uniqueness theorem for the singular envelope (Theorem 1.7). In fact we prove the existence of a canonical process which associates with each additive smooth-model categoryS a singular envelopeS-an ofS, whose objects are calledS-analytic spaces (Definition 5.1). We notice that most of the fundamental categories of geometry are of the formS-an (up to equivalence). As an application we introduce here two such categories: Banach differentiable spaces and Banach mixed spaces.
Similar content being viewed by others
References
Ceterchi, R.: A localization theorem for topological categories,Stud. Cerc. Mat. 42(1) (1990) 3–8.
Douady, A.: Le problème des modules pour les sous-espaces, analytiques d'un espace analytique donné,Ann. Inst. Fourier 16 (1966), 1–95.
Feit, P.: Axiomization of passage from “local” structure to global “object”,Mem. Amer. Math. Soc. 101(485) (1993).
Grothendieck, A.: Exposés 9–10 in Séminaire H. Cartan 1960/61 (Familles d'espaces complexes et fondements de la géométrie analytique).
Jurchescu, M.: Espaces mixtes, inSpringer Lect. Notes in Math. 1014, 1983, pp. 37–57.
Jurchescu, M.: Categories of abstract smooth models and geometric categories, 92/19, Preprint Dipartimento di Matematica “Guido Castelnuovo”, Roma.
Reichard, V.: Nicht differenzierbare morphismen differenzierbarer Räume,Manuscripta Math. 15 (1975), 243–250.
Serre, J.-P.: Faisceaux algébriques cohérents,Ann. of Math. 61 (1955), 197–278.
Spallek, K.: Differenzierbare Räume,Math. Ann. 180 (1969), 269–296.
Yoneda, N.: On the homology theory of modules,J. Fac. Sci. Tokyo 54 (1954), 193–227.
Author information
Authors and Affiliations
Additional information
The author is indebted to the Department of Mathematics of the University of Rome (La Sapienza) for hospitality and financial support and to the referees for their critical comments on the first version of this paper.
Rights and permissions
About this article
Cite this article
Jurchescu, M. Categories of abstract smooth models and their singular envelopes. Appl Categor Struct 3, 239–259 (1995). https://doi.org/10.1007/BF00878443
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00878443