# [FOM] Messing on isomorphism of categories

Colin McLarty colin.mclarty at case.edu
Wed Jan 9 13:13:33 EST 2008

```I have to point out that William Messing falls into a common
overstatement when he says:

> the notion of ismomorphism betweeen categories is (obviously) too
> strict for any serious mathematical use,

Much of the time it is true that we only care about a category up to
equivalence, or indeed only describe it up to equivalence.  But all
work with categories relies on isomorphisms of them.

As a trivial example:  Given categories A and B the product category
AxB is defined up to isomorphism, and not only equivalence, by the
familiar product property which we will use constantly.  The same holds
for the functor category B^A and all of the basic categorical
constructions.

A more advanced example is taken from Gelfand and Manin's book _Methods
of Homological Algebra_ -- even though that book itself is one of the
most noted sources of the idea that isomorphism of categories "appears
to be more or less useless" (p. 71).

Their central construction is the derived category D(A) of any Abelian
category A. Given A, they define D(A) by a universal property (p.
144).  Specifically D(A) is the universal target for a functor on
complexes on A which inverts quasi-isomorphisms.  This universal
property is used constantly, and all the uses depend on the fact that
it defines D(A) uniquely up to isomorphism of categories.

Most often in practice the Abelian category A will only be defined up
to equivalence in the first place -- and in this sense D(A) is also
defined only up to euqivalence.  But, *relative to the choice of A*,
the derived category D(A) is defined up to isomorphism of categories
and not only equivalence, and Gelfand and Manin's construction could
not be given without relying throughout on isomorphisms of categories.

best, Colin

```