(no subject)

[timelets]

So far, I see the following methods to construct a new domain from an existing one:

- product;
- co-product;
- free monoid;
- free graph;
- list.

Anything else?

http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1000858
(https://doi.org/10.1371/journal.pcbi.1000858)

Comments

[ecreet]

Domain as in "domain theory"? As in "integral domain"? or is it something else?

[timelets]

Good questions. It's in dom(f) when, e..g. I've got a bunch of objects and arrows and I want to construct a new domain. Does it make sense?

[ecreet]

More sense than before, but not quite enough :). What is f? If f is a morphism in a category, dom(f) is simply an object in this category. If f is a functor, dom(f) is a category.

When you say "domain", what kind of thing does this refer to? An object? A set of objects? A collection of objects and arrows? The constructions you list, what kind of thing each of them takes, and what kind of thing they produce? It seems to me that "free graph" in particular takes a set and produces an oriented graph.

Then, the most typical use of this word in math is "domain of a function" (more broadly, a morphism), and dom(f) suggests that this is the case. So, to have a domain we need to first have a function. When you say "construct a new domain", are you constructing a new function, too?

Perhaps you just mean "an object in a category", as in "a potential domain for a morphism in this category"? Then you are looking for ways to "construct" new objects "from" given objects? Do you care which category (i.e. sets?)?

[timelets]

I'm still having trouble formulating the right question. Maybe an example would help explaining what I mean. Consider Amazon's product rating system. Basically, every item in the their store is associated with a number of stars (1 to 5) and a number of customer reviews (0 to thousands). Based on a combination of these two numbers, people make purchasing decisions. To me, the combination is a new domain.

[ecreet]

Ok, so it seems that you have one "domain" - numbers from 1 to 5 - and another "domain" - non-negative integers, and from that you "make a new domain" - pairs of a number from 1 to 5 and a non-negative integer. In this example, your domains are sets, and the "new domain" is the direct product of two sets. It is still unclear what is f.

It looks like what you want is "having one, two, or more objects in a category, make another object in a (possibly different) category". Any functor (bifunctor, multifunctor) is going to give you this; those in your list are not special in this regard.

[The following is pure speculation and can be skipped.]
The "domain" terminology however suggests that you also want a map of morphisms, something like "if you have a map f originating from X and a map g originating from Y, you get a certain map originating from F(X,Y)". Then the question is what you want for targets of these maps. Again, any functor F(-,-) will give you "for f:X\to Z_1, g:Y\to Z_2 we have F(f,g):F(X,Y)\to F(Z_1,Z_2)". If however you want to keep the target always the same, you need a functor from a category to itself with a natural transformation to the identity functor, or a multifunctor with a natural transformation to the coproduct. For example, this is not happening for the direct product bifunctor on sets.

[timelets]

Thanks! Let me think about it a bit, esp. wrt the speculation part.

[timelets]

I really like the idea of a natural transformation to the coproduct. I'll try to diagram it and post a couple of figures.