On internal natural isomorphisms and topological categories

Consider internal categories and in a finitely complete ambient category , and functors . Assume we have a natural transformation . To say that is a(n internal) natural isomorphism, in my 2012 paper in TAC I wrote “We say a natural transformation is a natural isomorphism if it has an inverse with respect to vertical composition”, that is to say, there is some natural transformation such that the vertical composition and . For any internal category internal to a finitely complete category we can form the pullback (the sketchy formatting is deliberate here) Y^iso_1 ---> Y_1 x_{Y_0^2} Y_1 | | | | (m,m) | | v vY_0 x Y_0 ---> Y_1 x Y_1 u^2 where the top right pullback is Y_1 x_{Y_0^2} Y_1 ---> Y_1 | | | | (s,t) | | v v Y_1 -----------> Y_0 x Y_0 (t,s) Since is a section (as is a section of both and ) it is a monomorphism, and so the projection is a monomorphism. Moreover: CLAIM: is a monomorphism. Suppose I have such that . Write and , so that we have . Since , we can do so that , and…

Commenting on Bubbles has been removed. Why? If you want to talk about this post, write to the author, use the comments on their blog if it has them, or reply with a "RE:" post on your own blog. Suggestions? feedback@bubbles.town