Internal Categories in a Left Exact Cosimplicial Category

The notion of an internal category in a left exact cosimplicial category is introduced. For any topos over sets a certain left exact cosimplicial category is constructed functorially and the category of internal categories in it is investigated. The notion of a fundamental group is defined for toposes admitting the notion of “a discrete category.” Introduction Our primary interest in this paper is to introduce the notion of an internal category in a left exact cosimplicial category, generalizing ordinary internal categories. We call a cosimplicial category left exact when all of its components are categories with finite limits and all coface and codegeneracy functors preserve them. In Section 1 the definitions are given of an internal category in a category (which is wellknown) as well as in a left exact cosimplicial category. In Section 2 for any topos over Sets a certain left exact cosimplicial category is constructed functorially and the category of internal categories in it is investigated. Two examples of these constructions are considered. These examples correspond to the case where the topos is the category of sheaves over a locally compact topological space, and where it is a topos of presheaves. In Section 3 we consider toposes which admit the notion of “a discrete internal category.” For such toposes we determine the notion of fundamental group by means of the “discrete” category corresponding to the terminal object. When the topos is the category of sheaves over a locally compact and locally simply connected space, then its fundamental group is the same as the fundamental group of the underlying space. Some words about the notation: 1991 Mathematics Subject Classification. 76E15.