you can define $\Cat$ to be the 2-category of all $U'$-small categories, where $U'$ is some Grothendieck universe containing $U$. That way, you have $\Set \in \Cat$ without contradiction.

Do you agree with changing this to

” you can define $\Cat$ to be the 2-category of all $U'$-small categories, where $U'$ is some Grothendieck universe containing $U$. That way, for every small category $J$, you have the category $\Set^J$ an object of $\Cat$ without contradiction. This way, e.g. the diagram in Cat used in this definition of comma categories is defined. “

?

Reason: motivation is to have the pullback-definition of a comma category in (For others, it’s about the diagram here) defined, or rather, having Cat provide a way to make it precise. Currently, the diagrammatic definition can either be read formally, as a device to encode the usual definition of comma categories, or a reader can try to consult Cat in order to make it precise. Then they will first find only the usual definition of Cat having small objects only, which does not take care of the large category

$Set^I$

used in the pullback-definition. Then perhaps they will read all the way up to Grothendieck universes, but find that option not quite sufficient either since it only mentions Set, but not $Set^{Interval}$ . It seems to me that large small-presheaf-categories such as $Set^{Interval}$ can be accomodated, too, though.

(Incidentally, tried to find a “canonical” thread for the article “Cat”, by using the search, but to no avail. Therefore started this one.)

]]>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?

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