[FOM] Messing on isomorphism of categories

messing messi001 at umn.edu
Fri Jan 11 10:06:48 EST 2008

```Colin McLarty wrote:

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.
----------------------------------------------------------------------

It seems to me that it depends on one's point of view.  Notions of
category theory have a "strict sense" and a "loose sense".  If we define
the product of categories A and B in the strict sense, then, of course,
it is determined up to unique isomorphism by the requirement that for
any category C, Hom(C, A x B) ---> Hom(C, A) x Hom(C, B) be an
isomorphism, where Hom(?, ??) denotes the SET of functors with source ?
and target ??.  But if we denote by HOM(?, ??) the category of functors
with objects the functors from ? to ?? and morphisms the natural
transformations between such functors, then, sauf erreur, what we have
is an equivalence HOM(C, A x B) ---> HOM(C, A) x HOM(C, B).  Perhaps it
would be more convincing, if we take a functor F:I ---> Cat, where I is
a category and Cat is the category of U-categories, for some universe,
U, (or, if one prefers, the category of small categories).  Then there
is a category Lim(F), the inverse limit category whose objects are the
inverse limits of the family of sets Ob(F(i)), whose morphisms are the
obvious morphisms, equipped with  functors p_i:Lim(F) --> F(i), for all
objects i in I and satisfying, for w:i --> j a morphism in I, p_j =
F(w)p_i and which satisfies the universal property that, if C' is any
category and we are given a family of functors v_i:C' --> F(i), such
that if, w:i --> j is a morphism in I, we have v_j = F(w)v_i, then there
is an unique functor u:C' --> Lim(F) such that v_i = p_iu for all
objects i in I.  This is, defined up to isomorphism of categories.  On
the other hand, if we take not Cat, but CAT, the 2-category of
categories, where we regard not all natural transformations between
functors, but only the invertible natural transformations, then the
functor F:I --> Cat, has an inverse limit, LIM(F), but now only defined
up to equivalence of categories.  Objects of LIM(F) are families indexed
by the objects of I, (A_i), where A_i is an object of F(i), equipped
with, for each morphism w:i --> j in I, an invertible morphism
t_w:F(w)(A_i) --> A_j, such that, if w = Id_i, t_w = Id_{A_i} and if
w':j -->k is a second morphism in I, t_{w'w} = the composite
(t_w')(F(w')(t_w).  This concept of LIM(F) is utilized in any discussion
of fibered categories, stacks, gerbes, ....  It is discussed explicitly
in SGA 4, Giraud's book Cohomologie non-abelienne, the book by
Laumon-Moret Bailly Champs algebrique, ....  Perhaps its most common
occurence is in discussing the 2-fiber product of categories in
Cat_{/E}, that is the category whose objects are pairs (A,u) where A is
a category and u:A --> E is a functor.  Here the 2-fiber product of(A,u)
and (B,v) has objects (x, y, t) where x is an object of A, y is an
object of B and t is an isomorphism, t:u(x) --> v(y), in E.

William Messing
```