Equivalence of categories

Equivalence of categories

[Vasili I. Galchin]

> Dear Category Group,
>
>        What are some examples of invariants that are necessary for two
> categories to be equivalent?  E.g. If A and B are equivalent and A is a
> elementary topic, does this imply that B is an elementary topos?
>
> Regards,
>
> Vasily Gal'chin
>


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Re: Equivalence of categories

[Ingo Blechschmidt]

Dear Vasily,

On Sun, Feb 25, 2018 at 06:59:29PM +0000, Vasili I. Galchin wrote:
> What are some examples of invariants that are necessary for two
> categories to be equivalent?  E.g. If A and B are equivalent and A is a
> elementary topic, does this imply that B is an elementary topos?

yes, this is the case.

More generally, any property which can be formulated in the
first-order dependently-typed language of a category (consisting of a
type Ob of objects, a dependent type Hom(X,Y) for X, Y : Ob, an equality
predicate for values of Hom(X,Y), a fully-defined composition operation,
and the usual axioms) is invariant under equivalence of categories. Note
that, crucially, one cannot express equality of objects in this language.

For instance, the following properties are invariant under equivalence:

* There is an initial object.
* Any two parallel morphisms are equal.
* Any two endomorphisms of an object commute.
* There is an initial object, and any morphism into it is an isomorphism.
   (This property distinguishes the category of sets from its opposite
   category.)
* Any finite diagram has a limit.
* The category is abelian.
* The category is an elementary topos.

The following properties can not be formulated in the sketched language
and are indeed not invariant under equivalence:

* There is exactly one object.
* Isomorphic objects are equal.

Note that categories can fail to be equivalent even though they satisfy
the same statements. For instance, it's known that the theory of dense
linear orders without endpoints is complete. Therefore the thin
categories induced by rationals and by the reals satisfy the same
statements. But they are not equivalent (any equivalence is in fact an
isomorphism, and there is no bijection from Q to R).

Also note there are properties which are invariant under equivalence but
can't be expressed in the restricted language sketched above. For
instance:

* Any small diagram has a limit.
* The category is equivalent to the category of sheaves on a small site.

One would need an infinitary extension of the language to express these
properties.

Cheers,
Ingo

PS: The meta-claim "can be formulated in the language ==> invariant
under equivalence" holds constructively, without any appeal to the axiom
of choice. This is even so if you non-standardly define an equivalence
of categories to be an essentially surjective fully faithful functor. In
the absense of the axiom of choice, this definition is weaker than the
standard one (a pair of functors which are pseudo-inverse to each
other).

-- 
Ingo Blechschmidt
Lehrstuhl f??r Algebra und Zahlentheorie
Universit??t Augsburg
Tel.: 0821/598-5601


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