Dear Bill, Most constructivists would not have problems with this, despite the fact that at any finite stage of the construction you have only a finite approximation to the infinite category. Eventually (in finite time) you can get any finite part of the category, including the knowledge of any finite set of the equalities between objects or morphisms, and that is normally considered good enough. Computationally, a programming language such as Haskell is happy to deal with infinite objects. The computations never terminate, but every finite approximation to the object is obtained in finite time. For instance, one function might generate the decimal expansion of pi and another function may take that as input and calculate its square. In topos-valid constructivism the way it works is this. In an arbitrary elementary topos you can't always construct the free categories over graphs. But as soon as you admit a natural numbers object other free algebra constructions, such as the free category, come along with it. Regards, Steve Vickers. Galchin Vasili wrote:
Hello,
Let G be a directed graph that either has an infinite # of nodes or has edges which are loops.
Would a constructist recognize the existence of G's free category?
Thank you, Bill Halchin