1: How does the syllogism relate to the category?
Syllogism is a rule and system. Categories are produced by syllogism which is all the rules.

2: Topology where in Barr's text do you find something different than geometry?
(Page.2) ...the study of continuity: beginning with the continuity of space, or shapes, it generalize, and then by analogy leads into other kinds of continuity-and space as we usually understand it is left far behind.

3: where do we NOT use geometry in architecture?
Time, perception, space, change, continuity, distortion, stretching