1. How does the Syllogism relate to categories?

Aristotle defines syllogism as "a discourse in which, certain things having been supposed, something different from the things' supposed results of necessity because these things are so." Meaning if all animals are mortal and all humans are animals then all humans are mortal.

2. Where in Barr's text do you find something different than geometry?

Since topology is so difficult to define, it can have a different meaning than just being a kind of geometry, Barr states that "one might almost say it is a state of mind" and that it has it's own goal.

3. Where do we not use geometry in architecture?

In design we are always using geometry in one way or another. The only place I can think of where we are not using geometry is when we are arguing with clients about issues such as time schedules and budget, although those arguments are evidently about geometrical shapes of some sort.