Abstract
<p>The standard geometric realization functor (VBAR)?(VBAR)(,Y) : S('(DELTA)('op)) (--->) KTop, where S('(DELTA)('op)) is the category of simplicial sets and KTop is the category of compactly generated topological spaces, is generalized to the functor (VBAR)?(VBAR)(,Y) : B('(DELTA)('op)) (--->) A, where B is a topos, A is a category geometric over B via f, and g : Y (--->) f*(DELTA)('op) is a discrete fibration. The conditions under which this generalized functor preserves finite limits are sought. A related question of, for what categories E and internal categories C does the functor Colim(,C) : E('C) (--->) E preserves finite limits, is investigated first.For E belonging to a class of categories, referred to as admissible, which contains the toposes and also the category KTop, the functor Colim(,C) : E('C) (--->) E preserves finite limits if the internal category C is smooth, proper, and universal extremal filtered. If is further shown, for certain geometric categories A over sets, in particular for the categories Fco, ConsFco, Con, Lim, PsT, Born, and PreOrd, that initiality of the inclusion of the boundary (')Y(,on) of Y(,on) into Y(,on) guarantees that (VBAR)?(VBAR)(,Y) : S('(DELTA)('op)) (--->) A preserves finite limits. Moreover, in these cases the initiality condition may be viewed as a generalized Hausdorff condition on a quotient of the interval Y(,ol), since in the classical case, where A is the category KTop, it reduces to the Hausdorff condition.</p>