Re: the dual category

[Mamuka Jibladze <jib@rmi.ge>]

Alexander's example reminded me of something I always wanted to ask
somebody and never did, since it always felt too vague to me. But now I
thought - just ask.

In at least five very different contexts that I know, one seeks for a
nice placement of some very non-self-dual category against the
background of another one, "less non-self-dual".

In order of my increasing ignorance, these are:

Presenting spaces/locales/frames as certain (co/)monoids in the
category of sup-lattices, which is as nicely self-dual as it ever gets.

Extending the duality between discrete and compact abelian groups to
the self-dual category of locally compact abelian groups. There are
several closely related similar dualities, like e. g. the duality for
(locally?) linearly compact vector spaces by, I believe, Lefschetz. In
fact I think working with Banach or Hilbert spaces is largely motivated
by the desire to force infinite-dimensional vector spaces to behave more
like finite-dimensional ones, which form some of the nicest self-dual
categories.

Passing from (unstable) to stable homotopy theory is in a sense forcing
some amount of self-duality. The main feature of stable categories is
that they are additive (i. e. finite coproducts are isomorphic to the
corresponding products) but also much more - e. g. most of homotopy
cartesian or cocartesian squares in such categories turn out to be
homotopy bicartesian; this in particular implies the crucial feature
that the adjunction between suspension and loop space functors becomes
an equivalence (in a homotopy bicartesian square like

A -> 0
|    |
V    V
0 -> B

A is (stably equivalent to) the loop space of B iff B is (stably
equivalent to) the suspension of A; more generally, in a similar square

A -> 0
|    |
V    V
X -> B

A is the fibre of X -> B iff B is the cofibre of A -> X, etc.)

The context mentioned by Alexander, which triggered this post in the
first place - the phenomenon called limit-colimit coincidence: it seems
that imposing on some categories certain constructivity constraints
coming from computer science tends to imply certain amount of self-dual
features. Like, initial algebras for endofunctors become forced to
become isomorphic with final coalgebras for the same endofunctors. Or,
similarly, left adjoints to some functors to become isomorphic to right
adjoints to the same functors.

In physics, it seems that the main motivation of various quantization
procedures is to achieve certain amount of self-duality. For example,
evolution of a physical system becomes time-reversible.

It seems like in many cases such "self-dualization" can be formulated
in terms of forcing certain objects in certain monoidal categories to
become invertible but I don't know enough to tell more about it. In any
case I am aware of several works by category theorists which provide
appropriate formalism for such and similar constructions; the most
general formalism that I know is probably the Chu construction. But, if
I am not overlooking something obvious, I have only seen explanations of
*how* to "increase self-dual features", not *why* do these
self-dualization phenomena tend to occur in so many disparate contexts.

Does anybody know any underlying *reasons*? Can this phenomenon be
explained by the mere fact that "linearizing" the problem makes life
easier at the expense of losing certain amount of information, or there
actually exist some deeply rooted principles that force self-dual
behavior in certain mathematical or physical circumstances?

Sorry for this very vague and long post, but I am really eager to learn
about opinions of the category-theoretic community about this question
that I hardly ever managed to even formulate.

Mamuka


On Thu, 14 Sep 2017 15:53:21 +0100, Alexander Kurz <axhkrz@gmail.com>
wrote:
> I would like to add another example to Eduardo???s.
>
> In computer science both algebras and coalgebras for an endofunctor
> on sets are useful structures and both initial algebras and final
> coalgebras play an important role in the semantics of programming
> languages.
>
> It is now an important feature that algebras and coalgebras over set
> are not dual to each other. Only the invention of the dual category
> reveals the underlying duality.
>
> The ensuing tension between `abstract??? duality and `concrete???
> non-duality is certainly one reason why the study of set-coalgebras is
> fascinating.
>
> For example, whereas it is well-known that the initial sequence of a
> finitary set-endofunctor converges in omega steps, a result by Worrell
> shows that the final sequence of a finitary set-endofunctor converges
> in omega+omega steps.
>
> Best wishes, Alexander
>

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