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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 11th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexDeleted. Original content reproduced below: **Warning: This page is under construction. Feedback welcome.** See [[Understanding Constructions in Categories]] * [[Understanding Limits in Set]] * [[Understanding Colimits in Set]] *** ## Other Constructions ## #### Exponential Objects [[exponential objects]] #### Dependent Products [[dependent products]] #### Power Objects [[power objects]] ## Discussion ## +--{.query} [[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. =-- [[!redirects Understanding Set]] [[!redirects Understanding Constructions on Set]] [[!redirects understanding constructions in Set]] [[!redirects understanding Set]] [[!redirects understanding constructions on Set]] <a href="https://ncatlab.org/nlab/revision/diff/Understanding+Constructions+in+Set+%3E+history/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Understanding+Constructions+in+Set+%3E+history/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Understanding+Constructions+in+Set+%3E+history">current</a>

   Deleted. Original content reproduced below:

   Warning: This page is under construction. Feedback welcome.

   See Understanding Constructions in Categories

   • Understanding Limits in Set
   • Understanding Colimits in Set

   ---

   Other Constructions

   Exponential Objects

   exponential objects

   Dependent Products

   dependent products

   Power Objects

   power objects

   Discussion

   +–{.query} 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 . 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 is a cone over such that, given any cone , there is a unique cone morphism from to . So a univeral cone over the empty diagram is an object such that, given any object , there is a unique morphism from to . 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 is a terminal object, since there is a unique function (morphism) to from any set (object) . 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 be any function. Then

   for any element of , 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.

   =–

   !redirects Understanding Set !redirects Understanding Constructions on Set !redirects understanding constructions in Set !redirects understanding Set !redirects understanding constructions on Set

   diff, v15, current