Re: The boringness of the dual of exponential

[Uwe.Wolter@ii.uib.no]

Some short remarks from a dilettante in universal algebra, category
theory and philosophy:

I came along the unsatisfactory asymmetry in the category SET when
looking at coalgebras and asking about coequations some years ago.
Having in mind that kernels play a central role in universal algebra I
was expecting that co-kernels should play a similar role in universal
coalgebra. One can easily dualize the kernel reasoning. Such a
dualization, however, looks kind of artificial since a co-kernel just
provides a "strange coding" of the image of a map, i.e., a subset, so
to speak. One can identify the asymmetry by looking at exponentiation,
but isn't the asymmetry already related to the fact that SET is a
distributive category, i.e., that addition is trivial and boring
compared to multiplication?

My interest in coalgebras was also a kind of philosophically
triggered. The "classical western scientific cultur" relies and
focusses on the existence of "objects/things" and the category SET is
somehow the abstract essence of this perception of the world as a
bunch of objects/things which are assumed to be identifiable and
existing until the end of the time. Buddhistic and/or dialectical
reasoning, in contrast, perceives the world as a net of mutual
dependent and interweaved "processes". So, my question was if there is
anything in mathematics reflecting on a formal level this buddhistic
and dialectical reasoning based on "processes". I don't consider
coalgebras in SET as such formalization. Those coalgebras are based on
"object/thing reasoning" thus they can only give an approximation of
the philosophical concept of a "process", namely in terms of
distinctions and observations.

Uwe Wolter

Quoting Paul Taylor <pt11@PaulTaylor.EU>:

> When David originally posted his question, I thought it was rather
> a silly one and that it was quite rightly dismissed by various
> people.   On the other hand, he now says
>
>> However, I am not yet satisfied. Let me precise my thoughts. In the
>> textbooks and lecture notes on category category that I have read,
>> there are always product and coproduct, pullback and pushout,
>> equalizer and coequalizer, monomorphism and epimorphism, and so on.
>> However exponential is always left alone. That is why I assumed it is
>> boring. If it is not boring, why is it never mentioned in textbooks
>> and lecture notes on category theory?
>
> In other words, these things are "idioms" or "naturally occurring
> things" in mathematics, but there is a gap in the obvious symmetries.
>
> Looking for gaps in symmetries is a good thing to do.  For example
> Dirac (whose biography by Graham Farmelo I have just started reading)
> predicted the positron this way.
>
> Actually, if we're looking at the categorical structure of the category
> of sets, it isn't very symmetrical at all.  The second edition of
> Paul Cohn's "Universal Algebra" was evidently influenced by Mac Lane's
> famous textbook, but illustrates how categorists had way overemphasised
> duality.
>
> For example the terminal object yields the classical notion of element
> or point, whereas the initial object is strict and boring.
>
> Products and coproducts of sets are very different.
>
> I explored this kind of thing in my book.   For example, the section
> on coproducts shows how different they are in sets/spaces and algebras.
>
> So David's question becomes a good one that deserves an answer if
> we read it as one about the phenomenology of mathematics rather than
> its technicalities.
>
> Paul Taylor
>
> PS There is a boring technical answer that I don't think anyone has
> mentioned, namely copowers, especially of modules.
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]