1. How does syllogism relate to the categories?
Syllogism depends on Aristotle's categories by using systematic logic to determine relationships of things.
2. Where in Barr's text do you find something different than geometry?
Barr begins to describe topology as "a state of mind-and its own goal" and "space as we usually understand it is left far behind" once topology is understood. p2.
3. Where do we not use geometry in architecture?
Perhaps in virtual space when generating algorithms, but even so, the experience of these things are happening in a space dependent on geometry.