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Toposes and groupoids

1988; Springer Nature; Linguagem: Inglês

10.1007/bfb0081366

1617-9692

Ieke Moerdijk,

Microtubule and mitosis dynamics

The aim of this paper is to explain to what extent the category of Grothendieck toposes can be described in terms of groupoids (in the category of locales).In the first section, I describe how every groupoid G gives rise to a topos BG, and in section 2, I discuss some of the functorial properties of this constrution G ~ BG.After having introduced two completeness properties of groupoids, we will see that toposes can be obtained by localizing groupoids ( §4) or by considering geometric morphisms as obtained by tensoring with something analogous to a bimodule ( §5).In section 6 I briefly discuss the fundamental group of a topos.This paper provides a summary of my earlier papers [M2], [M3], [M4].Since the proofs given there are often long and technical, and involve extensive use of change-of-base methods, I believe it is worthwhile to present these results all together in a more directly accessible way, and save the reader from being distracted by perhaps less digestible technicalities.