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Hi,
I am working on a theory of physics that is intended to allow for variability over categories. By this I mean, a science that allows the user to reason over categories and even evolve his theory according to an evolution over categories. I intend for this theory of physics to allow for something called the “approximation and idealization of structure” and this is meant to allow a scientist to “have” an approximation to a structure which represents only the information which he has had access to up to some instant. The physics would allow the scientist to evolve his approximation and refine it given new information. This kind of mathematical method, I believe, would be relevant when considering a physics that is true at all stages in the history of an observer’s universe. For instance, in an early universe, when there are few or no events, the mathematical structure that is assumed to be relevant, only fits the data seen thus far. As a simple example, in a universe with only a single event, a theory should not presume more structure than would be exhibited by a system whose type is in a category containing only one morphism.
The example I want to talk about here is the approximation to a popular toy quantum category, $FREL$, the category of finite sets and relations. I believe this category is interesting to some modern researchers for two reasons. First, it is a toy quantum theory intended to piece apart quantumness by allowing only some quantum properties. Second, the category deals with finite sets and this has a flavour of quantum gravity to an extent. Regardless, $FREL$, along with $REL$ (sets and relations) are important toy quantum categories for present day researchers. In the spirit of the theory I am working on, we would reject either FREL and REL because it calls on all sets in their construction and neither the category of Sets or finite sets makes sense in a universe that has only a finite number of events. The solution to the problem is to have an approximation to $FREL$ that can evolve.
To approximate FREL we choose an ambient category, $Cat$, the category of small categories. $Cat$ is locally finitely presentable (which helps us). Next, we understand that the compact objects in Cat are the finite graphs. A finite set is a discrete category in $Cat$ and all finite sets are in $Cat$, namely the discrete (finite) categories. A relation between two finite sets is a finite graph. Take a first approximation to $FREL$, call it $APPR$, as a set of objects $ob(APPR) \in ob(Cat)$ and a set of morphisms $mor(APPR) \in ob(Cat)$. Next, consider adjoining an object and some morphisms to $APPR$ by finding the disjoint union of categories in $Cat$. Coproducts in $Cat$ are the disjoint union of categories. Here is a defintion from nlab and there is an example there for $Cat$.
Ultimately, we want to find $FREL$ as a colimit over all such categories in $Cat$.
How do we develop colimits in $Cat$ from coproducts and coequalizers? I have been told by a researcher at Oxford that coequalizers are very difficult in $Cat$, thus making colimits difficult. I am wondering if anyone can walk through this calculation with me?
I mention “adjoining” an object to $APPR$. By this I mean, take another category in $Cat$ that is similarly defined, so $APPR_2$ has the same kinds of objects and morphisms and $APPR$. To “adjoin” the new category, we find the coproduct, $APPR \sqcup APPR_2$. We want to build up such coproducts to have all the colimits for categories so defined and eventually to find $FREL$ as the colimit over all these. I don’t know how to do the colimit to fine $FREL$. Also, the researcher at Oxford, who I take on good authority, suggests that this method makes sense, but it should be a difficult calculation. I would like to do this calculation here on the nforum.
I didn’t mention that $APPR$ has, as objects, some set of finite discrete categories (ie, just some finite sets) and these are compact objects in $Cat$. The morphisms form some (finite) set of finite graphs (again, these are compact objects in $Cat$). The finite graphs, which form the morphisms, are the relations between the objects in the category $APPR$.
I think the mathematical part of this problem can be reformulated in the following two questions:
1). Is FRel a colimit of a diagram of finite discrete categories in Cat?;
2). Is FRel a colimit of a diagram of finite categories in Cat?
The answer to the first question is “no”, and the answer to the second question is “yes”.
The answer to the first question follows from the observation that Set may be regarded as a coreflective subcategory of Cat, thus any colimit of discrete categories in Cat is also discrete, but FRel is not discrete. The answer to the second question follows trom the fact that FRel is a colimit of the sequence FRel_1→FRel_2→FRel_3→… of finite categories FRel_n (full subcategory of FRel, whose objects are the first n natural numbers).
Actually I am not familiar with lfp-categories, but this n-lab page tells me that in lfp-category every object is a filtered colimit of the canonical diagram of compact objects (it is a part of the definition of lfp-category). So the question “How to construct a diagram of compact categories, which has FRel as a colimit?” is a part of the question “How to prove that Cat is a lfp-category”. In this case you should ask somebody familiar with lfp-categories for the reference.
That $Cat$ is lfp (locally finitely presentable) is well-known and follows from the fact that small categories are models of a finite limit sketch.
A really good source for all this is Adamek and Rosicky’s Locally Presentable and Accessible Categories.
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