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.
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