Why Arrow Worlds ?

Making a category out of a collection of transformations is unnecessary and requires an ugly shoe-horn.

The `natural transformation' is conventionally introduced into category theory as an adjunct of a pair of parallel functors, (F, G). The transformation, T, is defined to deliver a morphism of the destination category for each identity of the source category of the functors: if the first functor's image, F(f), of an arbitrary morphism, f, is composed with the transformation's image, Tω(f) of the identify, ω(f) at which that morphism ends, the composite, Tω(f) o F(f) is the same as that obtained from the transformation's image, Tα(f) of the identity, α(f), at which the morphism began and the second functor's image of the morphism, G(f): Tω(f) o F(f) = G(g) o Tα(f).

There is no reason why a single natural transformation (identified only in terms of its action on morphisms) need not transform between more than one pair of functors. Indeed, consider the simple category consisting of one non-identity morphism with identities at its distinct ends. A functor from this is described entirely by the morphism to which it takes the single non-identity. Then any commuting square, in any category, corresponds to a natural transformation; one pair of its sides correspond to the two functors, the other pair of sides are the transformation's images of the two identities and the diagonal is the common composite. If the transformation's image of the destination (source) identity is non-monic (non-epic) then it is trivial to construct two commuting squares with the same values for the natural transformation but a different functor on the left (right). [Draw it].

To identify a collection of natural transformations as a category we must be able to identify a single functor as the unique start-identity of any given transformation; likewise for its end. Clearly, then, we cannot say that a transformation is defined by its action on the identities of the source category (nor will we be saved by including the common composites as the transformation's action on non-identities): it also needs to be labelled with its start- and end-identities (any functor being self-conjoinable to deliver itself and so a natural transformation from itself to itself). Alternatively, we can abandon the notion of unique identities and allow there to be several things conjoinable before any given transformation, possibly all delivering the same conjoint.

In what follows, I propose to introduce category theory in terms of a more elementary construction in which transformations appear as the pre-eminent link between categories (or their precursors). That I have done this clumsily I am painfully aware; I shall be interested to see how it may be improved. It might be started rawer than it is; to what benefit I know not.

I have followed the school of thought which avoids reference to objects in category theory: identities can be persuaded to fill the rôle of objects (essentially by replacing any statement about a morphism's relationship with an object with an `equivalent' statement about its relationship with the identity on that object). This should be no surprise when I so clearly wish to de-emphasise (one might even say abandon) the rôle of the identity, itself: for this will be needed if we are to cope when, as discussed above, pre- and post-identities are not unique.

Naturally, of course, reducing to a lower layer of axiomatic decomposition also allows me to introduce definitions in terms of the properties they actually require, to explore the interaction of assorted definitions and to explore alternate axiomatisations of basic ideas. As I am committed to re-inventing notation, this exploration is simply a necessity.

Addendum: the virtue of Transformations not needing Functors.

First; a virtue of conjoining. Composition of transformations (including functors), unless they are auto, produces results not between the same arrow worlds as the composed transformations but between the source of one and the destination of the other. Yet conjoints act between the same arrow worlds as their factors. Indeed between any pair of arrow worlds, the collection of transformations forms an arrow world under conjoining. It looks pretty interesting to me. Cute idea: consider an arrow world, its dual, the arrow world conjoining transformations between them and an isomorphism (or possibly some other interesting kind of functor) between the last and the arrow world.

Eventually, I'll be wanting to describe some model of the universe in terms of some sort of arrow world (and, likely, categories will be involved). Quantum mechanics seems to need the ability to describe chains of events in terms of (densities and similar on) collections of possible histories (for example, each transition in some particle process is understood in terms of the Feynman diagrams for describable sequences of events effecting that transition). These histories proceed in steps, each of which has much the same character as the overall transition save that their intermediate `states' lack some of the character of definite reality.

A conjoinable list of transformations need not necessarily have any functors to compose between intermediates: even if our only understanding of meaning in our model is in terms of transformations between functors, it may help to describe these transformations as conjoints of lists. For instance, in describing a continuous progression of some physical process, we'd be very happy with a transformation describing the progress over some span (say of time) if, for any way of dividing that span in two, we could factorise the given transformation into a factor describing progress over the first part of the span conjoined with one describing progress over the second part. This may well encourage an interest in self-conjoinable transformations. In any case, such factorisations of transformations seems worth being discussable without needing to identify functors to serve as intermediate `identities' in the chain.

Suppose our model has functors describing (say) events (or maybe states understood by our model) and transformations describing the transitions between them. Factorising such a transformation as a conjoint of a list would amount to describing the transition in terms of various intervening sequences of `virtual' transitions. In so far as a factorisation may have functors composable between intermediates, we would have a (perhaps partial) understanding of the factorisation as expressing the effect of `going via an intermediate state'. Transformations do funky enough things that we can encode plenty of information in structures within the categories between which they transform: the transformation's action on the structures can be used to describe how the things described change. When considering transformations and their action on such structures within an arrow world, there may be much to be gleaned from `partial transformations', which only work on portions of the arrow world sufficient to contain such structures.

Histories have a quietly fractal characteristic: the passage of an electron and a positron `neglecting' to annihilate one another includes a plethora of histories in which they do annihilate, with the ensuing photon then spontaneously decaying back into the electrons; but the passage of a photon from one place to another equally involves its spontaneously turning into any manner of particle and its anti-particle, which then travel along for a bit and then annihilate, possibly doing plenty of more interesting things along the way; and the period between their formation out of the photon and their annihilation to recreate it is exactly the same as our initial electron and positron travelling a way and still both being there just before the annihilation.

Written by Eddy.
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