nLab
Understanding Constructions in Set

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See Understanding Constructions in Categories


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 SetSet. 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 JJ is a cone TT over JJ such that, given any cone CC, there is a unique cone morphism from CC to TT. So a univeral cone over the empty diagram is an object TT such that, given any object CC, there is a unique morphism from CC to TT. 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) CC. 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 !:C{!}'\colon C \to \bullet be any function. Then

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

for any element zz of CC, 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.

Revised on November 5, 2009 00:04:52 by Eric Forgy (65.163.59.49)