When a mathematician says that two categories are the same thing, they may mean there is an equivalence or an isomorphism between them. I am wondering if there is a precise way we can say that two categories that are isomorphic are different categories. I have no idea what I mean, but perhaps it has something to do with models of categories in Set. I just feel there may be justification sometimes when two mathematics feel that two isomorphic categories should be considered independently. Is this possible? Does it ever come up in discussion?
For example, consider this language on modules
"The concept of a Z-module agrees with the notion of an abelian groups."
This language "agrees" really means there is an isomorphism of categories. It seems like the theory of categories simply deletes any way of seeing them differently, but is there another way of seeing them where one might say"No, we consider them differently!"
I think this might have something to do with the way computer scientists talk about monads in a different way than mathematicians. If you've ever looked at these two literatures, they are nearly incompatible at times.
This may have something to do with the notion of "up to isomorphism" in category theory. When, in category theory, do we say, "no, I won't forget details that are deleted when we work 'up to isomorphism'"? Is there category theory that doesn't work "up to isomorphism"?
Which category theory concepts and constructions only work up to isomorphism and which, if any, go beyond this?
Best Answer
In an extremely literal Bourbakist sense, an abelian group is given by a pair $(A,\star)$ where $A$ is a set and $\star$ is a set of ordered pairs of the form $((a_1,a_2),a_3)$ such that all the $a_i$ are in $A$ and each element of $A\times A$ occurs as the first element of exactly one pair. Thus the category of abelian groups has as its class of objects a class all of whose elements are such pairs. (Of course not all such pairs form abelian groups, but given such a pair you can check whether it is an abelian group without providing any more structure.)
A $\mathbf Z$-module consists of a pair $((A,\star),\cdot)$ where $(A,\star)$ is as above and $\cdot$ is, to phrase it more colloquially than in the previous paragraph, a function $\mathbf Z\times A\to A$, again satisfying certain properties.
Thus the categories of abelian groups and of $\mathbf Z$-modules are certainly not equal: the class of pairs $(A,\star)$ and that of pairs $((A,\star),\cdot)$ are dramatically distinct. And we don't even need to go all the way to modules to get this kind of problem. For instance, why should it be a pair $(A,\star),$ rather than $(\star,A)$? Or perhaps I want a triple $(A,\star,e)$ rather than just an axiom saying that the identity $e$ exists? (Oh and by the way, have I defined ordered tuples so that $(A,\star,e)=((A,\star),e)$, or is it $(A,(\star,e)),$ or something else entirely?)
On the one hand, all of this is total nonsense and nobody cares; it is very hard to say that you're doing any kind of category theory if you're paying serious attention to the distinction between isomorphic but non-equal categories.
On the other hand, it does point to some non-nonsense (sense, I suppose?) as suggested in the other answer: rather than the rather old-fashioned and unmeaningful notion that we need to define an abelian group as some sort of ordered pair or triple, it is useful and important to consider that there are many different choices of syntax that you might use to compute in the theory of abelian groups. A very brief gloss of the idea of categorical semantics is that to give a syntax for abelian groups is to give a construction of the free category with finite products containing an abelian group object. All of the three-ish different syntaxes just mentioned give a distinct such construction of what is really the same category.