nLab Understanding Constructions in Set

Warning: This page is under construction. Feedback welcome.

See Understanding Constructions in Categories

• Understanding Limits in Set
• Understanding Colimits in Set

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Other Constructions

Exponential Objects

exponential objects

Dependent Products

dependent products

Power Objects

power objects

Discussion

Eric: How is this (note: terminal object) the universal cone over the empty diagram?

Toby: It seems to me that this is really a question about terminal objects in general than about terminal objects in $Set$. A cone over the empty diagram is simply an object, and a morphism of cones over the empty diagram is simply a morphism. A universal cone over a diagram $J$ is a cone $T$ over $J$ such that, given any cone $C$, there is a unique cone morphism from $C$ to $T$. So a univeral cone over the empty diagram is an object $T$ such that, given any object $C$, there is a unique morphism from $C$ to $T$. In other words, a universal cone over the empty diagram is a terminal object.

I don't see the point of the last paragraph before this query box. Already at the end of the previous paragraph, we've proved that $\bullet$ is a terminal object, since there is a unique function (morphism) to $\bullet$ from any set (object) $C$. It almost looks like you wrote that paragraph by modifying the paragraph that I had written in that place, but that paragraph did something different: it proved that ${!}$ was unique. Apparently, you thought that this was obvious, since you simply added the word ‘unique’ to the previous paragraph.

Alternatively, if you want to keep a paragraph that proves unicity, then you can remove ‘unique’ and rewrite my original unicity proof in terminology more like yours, as follows:

Now let ${!}'\colon C \to \bullet$ be any function. Then

$${!}'(z) = * = {!}(z)$$

for any element $z$ of $C$, so ${!}' = {!}$.

Eric: Thanks Toby. I think what I’m looking for is a way to understand that a singleton set is the universal cone over the empty diagram. All these items should be seen as special cases of limit. Unfortunately, I don’t understand limit well enough to explain it. In fact, that is one of the reasons to create this page, i.e. so that I can understand limits :)

The preceding paragraph was my attempt to make it look like a limit, but I obviously failed :)

Ideally, this section would show how terminal object is a special case of limit somehow.