At the basic level, a category consists of objects and arrows (or even with just arrows), and identity is required only for arrows. Objects are only important upto isomorphism. A general guideline I follow for modelling some phenomenon witht categories is to try and map concepts to objects and arrows based on whether the concept requires identity or isomorphism. If identity does not seem to appear anywhere, then we need to consider using higher-order categories such as functor categories. However it is important to drill down all the levels of the categories and identify the concepts for which identity is important and model these concepts using arrows. Of course, this is easier said than done, and we need considerable practice in working out all the book-keeping and figuring out all the universal properties for all the categories involved, before we can start performing "magic" with "abstract nonsense".
Anyway, back to my original topic of Vaughan Pratt's post. Let me reproduce the post verbatim so that I do not lose the original trigger to my thoughts:
Date: Sat, 18 Apr 2009 17:14:51 -0700From: Vaughan PrattSubject: [FOM] When is it appropriate to treat isomorphism as identity?To: Foundations of MathematicsMessage-ID: <49ea6cfb.7090802@cs.stanford.edu>Content-Type: text/plain; charset=ISO-8859-1; format=flowedMy recent complaint that uniqueness is too often taken for granted when dealing with questions of existence brought to light the ease with which people identify isomorphism with identity: a number of people wrote to remind me that uniqueness is common in many mathematical contexts, e.g. the cyclic group of order 5, the least upper bound of two elements of a lattice, etc.Operations of course enforce uniqueness by definition of "operation": the sum of two integers, the concatenation of two lists, the union of two sets, etc. However in a good many of the examples, "the" was not enforced by some operation but instead only meant "up to isomorphism."Sometimes it is necessary to treat isomorphism as identity, sometimes it is merely convenient, sometimes it is inconvenient, and sometimes it is inconsistent or paradoxical. Here are examples of each.====================1. Necessary(a) Equivalence definitions of "lattice". A lattice can be defined as a poset every pair of elements of which has a least upper bound and a greatest lower bound, or as an algebra with operations of meet and join satisfying the lattice laws (idempotence, commutativity, associativity, absorption). A poset or partially ordered set is a set with a reflexive transitive antisymmetric binary relation. The antisymmetry requirement, that (in essence) isomorphic elements are equal, is *necessary for these two definitions to be equivalent*. The notions of least upper bound and greatest lower bound continue to be meaningful for preordered sets, those for which the relation is only required to be reflexive and transitive, but preordered sets with all binary lubs and glbs do notform a natural class of algebras.(b) Birkhoff duality of finite posets and distributive lattices, as per David Eppstein's excellent article athttp://en.wikipedia.org/wiki/Birkhoff's_representation_theorem#Functoriality. The monotone functions from a finite poset P to the two-element chain, i.e. its order filters, form a finite distributive lattice P*, the filters (homomorphisms to the two-element lattice) of which recover P up to isomorphism. No such duality obtains for preordered sets, making antisymmetry necessary for Birkhoff duality. Actually there are two types of isomorphism entering here, the isomorphism of elements in a clique within a preordered set, and the isomorphism between P and the poset of lattice filters of P*; regarding the latter it is convenient to think of the two posets as the same, but set theoretically unsound, seebelow under "4. Inconsistent".(c) Enumeration of structures. Neil Sloane's Encyclopedia of Integer Sequences counts the number A_0, A_1, A_2, ..., A_n, ... of posets, distributive lattices, tournaments, and many other structures of a given size n. This only makes sense for structures defined up to isomorphism, as otherwise every A_i would be a proper class. (There is however anotion of labeled structure which permits enumeration by assuming in effect that the n elements are the numbers from 1 to n; with that convention there is one labeled set of each finite cardinality, but three labeled posets with two elements.)2. ConvenientWe already mentioned the convenience of identifying P with the filters of P* above. While it's not necessary to identify every countable dense linear order with the unit interval of rationals (open, half-open, or closed according to which endpoints exist), it can be convenient. Thesame goes for a great many other structures, e.g. the free group on one generator as the group of integers, the initial ring as the ring of integers, etc. Sometimes there is no canonical object, for example the Boolean algebra of finite and cofinite subsets of the set N of natural numbers under union, intersection, and complement relative to N is isomorphic to that of the ultimately constant binary sequences under coordinate-wise Boolean combination, with no clear advantage to either.The same goes for the Boolean algebra of periodic binary sequences, which is isomorphic to the free Boolean algebra on N as well as to the (unique) countably infinite atomless Boolean algebra.3. Inconvenient.The forgetful functor U: Pos --> Set producing the underlying set of each poset in the category Pos has a left adjoint F: Set --> Pos which maps each set X to the corresponding discrete poset. Inconveniently, U has no right adjoint. On the other hand the category Ord of preordered sets, which extends Pos, has a forgetful functor U+: Ord --> Set which conveniently has both a left and a right adjoint. The left adjoint F+: Set --> Ord yields discrete preordered sets as for Pos, while the right adjoint G: Set --> Ord yields codiscrete preordered sets or cliques, which don't exist in Pos for posets with two or more elements.4. InconsistentThe powers N, N^2, N^3, N^4 of the set N of natural numbers are all isomorphic. Set theory promises that we can implement N^3 as either Nx(NxN) or (NxN)xN. If we use the former then the three projections are p, pq, and q^2, if the latter then p^2, qp, and q. This works fine as long as Nx(NxN) and (NxN)xN remain distinct (albeit isomorphic) sets. However if we identify NxN with N then according to set theory we must have p = p^2, pq = qp, and q^2 = q. But then p, pq, and q are the only functions that can be formed from the two projections from NxN, making it impossible to obtain four distinct projections from N^4 no matter how realized using NxN.To make things consistent we could disallow the implementation of N^3 as (NxN)xN, but then it would no longer be set theory as we know it, but some other theory of sets that we would have to work out afresh. Identifying isomorphic sets is inconsistent with set theory asstandardly defined and used.(This adapts to set theory a similar argument by John Isbell for monoidal categories, see Mac Lane, Categories for the Working Mathematician, near the end of VII-1.)=====================A benefit of mathematical logic is that it makes explicit principles of logic that we might otherwise consider inviolable, allowing us to consider the consequences of violating them by replacing them with competing rules, thereby contributing to the ongoing undermining of the efficacy of pure reason.This raises the following question. It is clear that arguments purporting to show either the existence or uniqueness of any given entity beg the question of the necessity of the rules used in those arguments. That said, is the situation entirely symmetric between existence and uniqueness, or is there some reason to suppose that existence arguments might be supportable by rules having a more necessary character than those supporting uniqueness arguments? To demonstrate such an asymmetry would not require absolutely necessary rules, only some generally accepted or at least plausible ranking of rules by perceived necessity.Vaughan Pratt
