Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Simple Questions about Category of Sets

Julius Hamilton (Jul 02 2024 at 20:47):

I would love if someone could discuss this with me.

For a model of financial trading, I have the following sets:

• the set of all people
• the set of all assets (for example, US dollars, Chinese Yen, etc.)
• the set of all moments in time
• the set of all quantities

We can express who owns how much of what and when, with a relation: $Owns \subseteq People \times Assets \times Quantity \times Time$.

However, if it would be easier to have temporal intervals, I think the power set of Time reflects the set of all possible temporal intervals or collections of temporal moments.

Does anyone think it would make more sense to define Intervals as I think "connected" sets, in this context? (I mean continuous intervals like $[10, 20]$ and not unions of subsets like $\{0\} \cup [0.5, 1]$). (It's worth considering that every currency has a finite quantity in circulation, and any unit of that currency cannot be owned by two people at the same time, in this model, for now.)

Does that set of temporal intervals have a universal property? I mean if there is a categorical relationship between a set of temporal moments and a set of temporal intervals. My guess is that the latter is a topology on the former, so I wonder if there is a "categorical" or "internal" definition of a topology on an object.

I am also thinking about the universal property of the power set. I'm gonna have to process this:
exponentialObject.png

I am also wondering, if I design a satisfactory model of the above in the category of sets, and recognize how the sets relate to each other categorically, are there benefits to swapping the underlying category but keeping the same categorical structure? Rumy Tuyeras suggested I work in a category of monoids. What happens if I design the model in the category of sets, identify the universal properties of the objects, and then switch the underlying category to the category of monoids?

Thanks.

Julius Hamilton (Jul 03 2024 at 15:21):

Let me attempt to process the diagram of the universal property of exponential objects.

exponentialObject.png

$X$ is a variable ("For any object $X$ in $C$...").

$Y$ is a "distinguished object", in the sense that we require it have certain properties, for the rest of the definition to follow; which is that $C$ includes all binary products "with $Y$". I think this means both $X \times Y$ and $Y \times X$.

I wonder what the significance is of a category with all binary products for a single object, rather than for all objects.

$Z$ is a specified object, but the definition does not impose any required properties on it (I mean, prior to its role in the rest of the definition).

$Z^Y$ is a specified object, which is required to have a morphism $Z^Y \times Y \to Z$. Verbally, this says that for the product of $Z^Y$ and $Y$ (which we know exists from the stated property of $Y$, above), there must exist a morphism from that, to $Z$. So, so far, we are considering a situation like this:

diagram1.jpeg

For $Z^Y$ to be an exponential object:

For any object $X$, we know $X \times Y$ exists. I do not think a morphism $X \times Y \to Z$ has to exist. If it does: there must be a unique morphism $X \to Z^Y$, which (very roughly speaking) makes makes morphism $Z^Y \times Y \to Z$ equal to morphism $X \times Y \to Z$ (when composed). In other words (I think):

In this situation:

diagram2.jpeg

The red line indicates "we want to be able to get from here to here":

diagram3.jpeg

My current idea for (approximately) distilling this into a simple concept is, "We want there to be a correspondence between a product of two objects and some other object, where we can sort of "factor out" one of the objects ($Y$), and express the relationship as a single arrow between an object ($Z^Y$) and $Z$."

I think that this can also be understood in the reverse direction, where we can conceptualize of the relationship between an exponential object $Z^Y$ and its "base object" $Z$, as a particular relationship directly between $Y$ and $Z$.

This is all a work in progress, so no conviction in anything I've said. :+1:

(I missed the $X \to Z^Y$ morphism in my drawing, whoops.)

Or maybe this is a simple way to put it:

If there is some "exponential object" $E$, the relationship between every other object in the category, say, $X$, and $E$, can be re-expressed or maybe "decomposed", as a relationship between $X \times Y$ and $E \times Y$ - it's as if there is some "additional hidden object" we add in (I think it appears to be). If this is the case, we can rewrite $E$ to express almost how it is "controlled" by $Y$ - $E$ is actually a relationship between some object $Z$ and $Y$ (I think).