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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeFeb 24th 2010
- (edited Jan 23rd 2013)
- PermaLink
Author: Urs
Format: Markdownadded to [[Set]] the statement that <latex>Set</latex> is the terminal topos.

added to Set the statement that $Set$ is the terminal topos.
- CommentRowNumber2.
- CommentAuthorTodd_Trimble
- CommentTimeFeb 24th 2010
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Author: Todd_Trimble
Format: MarkdownI added the modifier 'Grothendieck' to topos, since for example there is no geometric morphism <latex>FinSet \to Set</latex>.

I added the modifier 'Grothendieck' to topos, since for example there is no geometric morphism $FinSet \to Set$ .
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeFeb 24th 2010
- PermaLink
Author: Urs
Format: MarkdownAh, right. Thanks.

Ah, right. Thanks.
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeFeb 24th 2010
- PermaLink
Author: Urs
Format: Markdownhm, but then I have a mistake in the section <a href="http://ncatlab.org/nlab/show/global+section#GeneralTopos">global section functor on a general topos</a>...

hm, but then I have a mistake in the section global section functor on a general topos...
- CommentRowNumber5.
- CommentAuthorMike Shulman
- CommentTimeFeb 24th 2010
- PermaLink
Author: Mike Shulman
Format: MarkdownWhy is the left adjoint in that section called "LConst" rather than "Const"?

Why is the left adjoint in that section called "LConst" rather than "Const"?
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeFeb 24th 2010
- PermaLink
Author: Urs
Format: Markdown> Why is the left adjoint in that section called "LConst" rather than "Const"? For two reasons: 1. <latex> LConst_S</latex> is the constant sheaf of locally constant functions! The sheaf is named after what it is "a sheaf of". 2. for <latex> C</latex> a site, the functor <latex> Set \to PSh(C) </latex> should be called <latex> Const </latex> . Then with <latex> Sh(C) \stackrel{\stackrel{L}{\leftarrow}}{\stackrel{R}{\to}} PSh(C) </latex> the geometric embedding, we have the pleasant <latex> LConst = L \circ Const </latex>
Why is the left adjoint in that section called "LConst" rather than "Const"?

For two reasons:
1. $LConst_S$ is the constant sheaf of locally constant functions! The sheaf is named after what it is "a sheaf of".
2. for $C$ a site, the functor $Set \to PSh(C)$ should be called $Const$ . Then with $Sh(C) \stackrel{\stackrel{L}{\leftarrow}}{\stackrel{R}{\to}} PSh(C)$ the geometric embedding, we have the pleasant $LConst = L \circ Const$
- CommentRowNumber7.
- CommentAuthorMike Shulman
- CommentTimeFeb 24th 2010
- PermaLink
Author: Mike Shulman
Format: MarkdownHmm, I guess I see that, but my intuition is that Const(X) should be the constant sheaf on X.

Hmm, I guess I see that, but my intuition is that Const(X) should be the constant sheaf on X.
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeFeb 24th 2010
- (edited Feb 24th 2010)
- PermaLink
Author: Urs
Format: Markdown> <latex>Const(X)</latex> should be the constant sheaf on <latex>X</latex>. Well, it's called the "constant sheaf" of course. I still found it a great idea to write <latex>LConst_S</latex> for the constant sheaf on <latex> S </latex>. Originally I kept writing <latex> L Const </latex> with a little space to amplify that it is the sheafification of the constant presheaf. Then it seemed like a neat idea to just discard the space. I think it fits nicely into the pattern. This way for instance we have precisely that <latex> \Gamma(LConst_{Set}) = LConst(X)</latex> is the set of locally constant functions. More generally, for <latex> LConst_{Core((n-1)Grpd)}</latex> the constant <latex>n</latex>-stack on the n-groupoid of (n-1)-groupoids, we have <latex> LConst(X) = \Gamma(LConst_{Core((n-1)Grpd)})</latex> is precisely the n-groupoid of locally constant (n-1)-stacks. So that works very nicely. If I write <latex> Const </latex> instead of <latex> LConst </latex> here a very nice abstract pattern is severely broken by the notation. Or so I think.

$Const(X)$ should be the constant sheaf on $X$ .

Well, it's called the "constant sheaf" of course. I still found it a great idea to write $LConst_S$ for the constant sheaf on $S$ . Originally I kept writing $L Const$ with a little space to amplify that it is the sheafification of the constant presheaf. Then it seemed like a neat idea to just discard the space.

I think it fits nicely into the pattern. This way for instance we have precisely that $\Gamma(LConst_{Set}) = LConst(X)$ is the set of locally constant functions.

More generally, for $LConst_{Core((n-1)Grpd)}$ the constant $n$ -stack on the n-groupoid of (n-1)-groupoids, we have
$LConst(X) = \Gamma(LConst_{Core((n-1)Grpd)})$
is precisely the n-groupoid of locally constant (n-1)-stacks. So that works very nicely. If I write $Const$ instead of $LConst$ here a very nice abstract pattern is severely broken by the notation.

Or so I think.
- CommentRowNumber9.
- CommentAuthorMike Shulman
- CommentTimeFeb 25th 2010
- PermaLink
Author: Mike Shulman
Format: MarkdownI just find it confusing to write "LConst" for the functor which constructs constant sheaves -- my mind keeps trying to read it as a functor which constructs locally constant sheaves. In general I think it's more common to name functors after what they do to objects, not to the elements of those objects. For instance, we say "the free monoid functor" not "the words functor," and "the free strict oo-category functor" not "the pasting diagrams functor." Probably you agree that the functor is "the constant sheaf functor," but then I think the notation of the functor should reflect the name of the functor.

I just find it confusing to write "LConst" for the functor which constructs constant sheaves -- my mind keeps trying to read it as a functor which constructs locally constant sheaves. In general I think it's more common to name functors after what they do to objects, not to the elements of those objects. For instance, we say "the free monoid functor" not "the words functor," and "the free strict oo-category functor" not "the pasting diagrams functor." Probably you agree that the functor is "the constant sheaf functor," but then I think the notation of the functor should reflect the name of the functor.
- CommentRowNumber10.
- CommentAuthorDavidRoberts
- CommentTimeFeb 25th 2010
- PermaLink
Author: DavidRoberts
Format: MarkdownI think there are some level-type conflations here. The constant stack with fibre the groupoid Set is the stack of locally constant sheaves. The constant sheaf with fibre F is the sheaf of locally constant functions with values in F.

I think there are some level-type conflations here. The constant stack with fibre the groupoid Set is the stack of locally constant sheaves. The constant sheaf with fibre F is the sheaf of locally constant functions with values in F.
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeFeb 25th 2010
- PermaLink
Author: Urs
Format: Markdown> In general I think it's more common to name functors after what they do to objects, not to the elements of those objects Here is an idea: would it help if I write <latex> LConst(-)</latex> systematically?

In general I think it's more common to name functors after what they do to objects, not to the elements of those objects

Here is an idea: would it help if I write $LConst(-)$ systematically?
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeJan 23rd 2013
- PermaLink
Author: Urs
Format: Markdownadded to _[[Set]]_ a brief paragraph _[Set -- Properties -- Opposite category and Boolean algebras](http://ncatlab.org/nlab/show/Set#OppositeCategory)_

added to Set a brief paragraph Set -- Properties -- Opposite category and Boolean algebras
- CommentRowNumber13.
- CommentAuthorTodd_Trimble
- CommentTimeJan 23rd 2013
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexI sharpened Urs's paragraph. Incidentally, I didn't understand why this was cast in a definition environment (definition 1).

I sharpened Urs’s paragraph. Incidentally, I didn’t understand why this was cast in a definition environment (definition 1).
- CommentRowNumber14.
- CommentAuthorUrs
- CommentTimeJan 23rd 2013
- PermaLink
Author: Urs
Format: MarkdownThanks! The Definition-environment was supposed to be a Proposition-environment. I have changed it now. (This happens because I grab the code for these environments from the Instiki-Theorems page where it is the Definition-environment that is written out. Here I forgot to edit it after copy-and-pasting.)

Thanks!

The Definition-environment was supposed to be a Proposition-environment. I have changed it now. (This happens because I grab the code for these environments from the Instiki-Theorems page where it is the Definition-environment that is written out. Here I forgot to edit it after copy-and-pasting.)
- CommentRowNumber15.
- CommentAuthorIngoBlechschmidt
- CommentTimeApr 11th 2014
- PermaLink
Author: IngoBlechschmidt
Format: MarkdownItexAdded to _[[Set]]_ the following remark: "In constructive mathematics, $\mathcal{P}$ defines an equivalence of $\Set$ with the opposite category of that of [[complete lattice|complete]] [[atom|atomic]] [[Heyting algebras]]. In fact, for any elementary [[topos]] $\mathcal{E}$, the [[power object]] functor defines an equivalence of $\mathcal{E}$ with the opposite category of that of internal complete atomic Heyting algebras. (This phrase can be interpreted using the [[internal language]] of $\mathcal{E}$.)" Note that at _[[complete atomic Boolean algebra]]_, there is the following very interesting statement: "This property of CABAs is not applicable in [[constructive mathematics]], where power sets are rarely boolean algebras. However, we can use [[discrete space|discrete]] [[locales]] instead (or rather, their corresponding [[frames]]). That is, define a __CABA__ to be (not a complete atomic boolean algebra but) a frame $X$ such that the locale maps $X \to 1$ and $X \to X \times X$ (which in the category of frames are maps $0 \to X$ and $X + X \to X$) are [[open map|open]] (as locale maps). Then it should be (I will check) a classical theorem that CABAs and complete atomic boolean algebras are the same, and a constructive theorem that CABAs and power sets are the same (in the same functorial manner as above)." This is should really be checked someday. I will do it, but not now.

Added to Set the following remark:

“In constructive mathematics, $\mathcal{P}$ defines an equivalence of $\Set$ with the opposite category of that of complete atomic Heyting algebras. In fact, for any elementary topos $\mathcal{E}$ , the power object functor defines an equivalence of $\mathcal{E}$ with the opposite category of that of internal complete atomic Heyting algebras. (This phrase can be interpreted using the internal language of $\mathcal{E}$ .)”

Note that at complete atomic Boolean algebra, there is the following very interesting statement:

“This property of CABAs is not applicable in constructive mathematics, where power sets are rarely boolean algebras. However, we can use discrete locales instead (or rather, their corresponding frames). That is, define a CABA to be (not a complete atomic boolean algebra but) a frame $X$ such that the locale maps $X \to 1$ and $X \to X \times X$ (which in the category of frames are maps $0 \to X$ and $X + X \to X$ ) are open (as locale maps). Then it should be (I will check) a classical theorem that CABAs and complete atomic boolean algebras are the same, and a constructive theorem that CABAs and power sets are the same (in the same functorial manner as above).”

This is should really be checked someday. I will do it, but not now.
- CommentRowNumber16.
- CommentAuthorUrs
- CommentTimeMay 20th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Saunders MacLane]], §IV.10 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)] <a href="https://ncatlab.org/nlab/revision/diff/Set/39">diff</a>, <a href="https://ncatlab.org/nlab/revision/Set/39">v39</a>, <a href="https://ncatlab.org/nlab/show/Set">current</a>
added pointer to:
- Saunders MacLane, §IV.10 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
diff, v39, current
- CommentRowNumber17.
- CommentAuthorvarkor
- CommentTimeJan 3rd 2024
- PermaLink
Author: varkor
Format: MarkdownItexWhat does the phrasing: >As a groupoid, $Set$ is [...] mean on this page? Is it referring to the groupoid of sets and bijections?

What does the phrasing:

As a groupoid, $Set$ is […]

mean on this page? Is it referring to the groupoid of sets and bijections?
- CommentRowNumber18.
- CommentAuthorUrs
- CommentTimeJan 3rd 2024
- PermaLink
Author: Urs
Format: MarkdownItexThis paragraph seems to originate in [revision 32](https://ncatlab.org/nlab/revision/diff/Set/32) signed by "Anonymous" in Feb 2021. But yes, it must be referring to the core groupoid. I have made a little edit to clarify. <a href="https://ncatlab.org/nlab/revision/diff/Set/42">diff</a>, <a href="https://ncatlab.org/nlab/revision/Set/42">v42</a>, <a href="https://ncatlab.org/nlab/show/Set">current</a>

This paragraph seems to originate in revision 32 signed by “Anonymous” in Feb 2021.

But yes, it must be referring to the core groupoid. I have made a little edit to clarify.

diff, v42, current

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