Can someone give me an example of a category (telling me what the objects and morphisms are) where the objects are not some construction on sets or classes?
closed as not a real question by Andreas Blass, David White, Fernando Muro, George Lowther, Todd Trimble♦ Dec 17 '12 at 22:31It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 


The free topos (rather over the top, but still). See the book of Lambek and Scott. In categorical logic and the computer science associated to functional programming, the issue is really the other way round. There a category is often given via "syntax" and finding some model in which the objects are sets with possible structure is one of the basic aims of the theory. (If morphisms are some sort of program, or proof, objects are not so transparent.) 


You say: where the objects are not some construction on sets or classes. What do you actually mean here? I will give several examples of my interpretation of this, in some you may think it violates that criterion.
(Just in case the question resulted from a problem set in a category theory course, I have left you to write down ALL the details! In any case, it is a good idea to do so.) 


I do not know exactly what is asked. For me all objects of Mathematics are sets. But perhaps what you want is what I used as an example to illustrate the concepts of Category Theory. (I call it the Toy Category). Objects = Natural numbers. Morphism: $\alpha\colon n\to m$ such that $m\ \vert\ \alpha \gcd(n,m)$ (here $\alpha\in{\bf Z}$). Two morphisms $\alpha\colon n\to m$ and $\beta\colon n\to m$ are considered the same if $\alpha\equiv \beta\bmod m$. For example $35\colon36\to60$ is a morphism. The composition of two morphisms $\alpha\colon n\to k$ and $\beta\colon k\to m$ is by definition $\alpha\beta\colon n\to m$. You may check it is well defined ( and the compositions of equivalents gives equivalents ). You may check this is a Category, and for example every isomorphism is an automorphism and the Group of automorphisms of n is isomorph to $({\bf Z}/n{\bf Z})^*$. Is this what you want? 


Category theory does not base on sets. More, there's no such thing as "the" Set Theory. Each theory has its own collection of axioms. If your specific category does not involve any particular set theory, it does not rely on set theory. You may think of modeling categories in certain set theories; but again, it depends on how you define sets. Say, you can (kind of frivolously) define sets as discrete categories. 

