Dana Scott said the other day, in a presentation, "Haskell Curry was the only Mathematician I ever knew who could see a concept only in terms of symbols." I think that's a rather elegant compliment to Curry, and conversely, I think it says a lot about how we think about Mathematics, since we are not all Haskell Curry. I'm venturing that a lot of us expose our ideas systematically, but, in the back end, we have our own devices, which let us understand difficult concepts, and render new insight through their manipulation. I'm curious what sort of devices we have. I know that some purists are probably shifting uneasily at this. However, I wonder why we call the application of a functor a "lifting", and why a fibration F: E -> C suggests that "E lies over C", and we are all very happy to draw a little loop over an object to denote a fiber. This comes from somewhere. Why do we call epis and monos "split" if they have an inverse? Why is the pseudo-inverse of a fibration called a "cleavage"? I literally picture a bundle of fibers getting cleaved by an axe-like object in my mind's eye. I'm actually fairly new to Category Theory, and I'm wary of mental models that would take me down the wrong path, or deprive me of the elegance of a fully symbolic understanding. However, I don't aspire to be great at Mathematics. I'm just not that good. It would be wonderful if I could share in some mental helpers that would give me insight into the topics I'm studying. - Brian [For admin and other information see: http://www.mta.ca/~cat-dist/ ]