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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 12th 2010
   • (edited Oct 12th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexDave Carchedi observes the following about the topos of sheaves on [[CartSp]]: It is acually a _local topos_ in that the terminal geometric morphism is not only essential, but there is a fourth adjoint $Sh(CartSp) \stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}} Set$, where for $S$ a set, $Codisc S$ is the codiscrete smooth structure on $S$ in that for $U \in CartSp$ we have $$ Hom_{Sh}(U, Codisc S) \simeq Hom_{Set}(\Gamma(U), S) \,, $$ where on the right we have the set of maps from the underlying set of the Cartesian space $U$ into the set $S$. Now the observaton is that the unit of this adjunction $$ X \to Codisc \Gamma X $$ sends the set of $U$-plots of $X$ into the set of set-maps $Hom_{Sh(CartSp)}(U,X) \mapsto Hom_{Set}(\Gamma(U),\Gamma(X)) $. This is noteworthy, because the image-factorization of this map is _concretization_ the map that sends a sheaf to its underlying [[concrete sheaf]]: the underlying [[diffeologial space]]. So we are wondering if this points to a nice general abstract way of understanding how the quasi-topos of concrete sheaves sits inside the topos of all sheaves. Is there any good way to speak for a local topos of the image factorization of the unit map of the extra adjunction? And generally: is there known for quasi-toposes any characterization akin to the characterization of sheaf toposes as accessible exact reflective subcategories of presheaf toposes?

   Dave Carchedi observes the following about the topos of sheaves on CartSp:

   It is acually a local topos in that the terminal geometric morphism is not only essential, but there is a fourth adjoint $\mathrm{Sh}(\mathrm{CartSp})\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{Codisc}}{\leftarrow }}\mathrm{Set}$,

   where for $S$ a set, $\mathrm{Codisc}S$ is the codiscrete smooth structure on $S$ in that for $U\in \mathrm{CartSp}$ we have

   $${\mathrm{Hom}}_{\mathrm{Sh}}(U,\mathrm{Codisc}S)\simeq {\mathrm{Hom}}_{\mathrm{Set}}(\Gamma (U),S),$$

   where on the right we have the set of maps from the underlying set of the Cartesian space $U$ into the set $S$.

   Now the observaton is that the unit of this adjunction

   $$X\to \mathrm{Codisc}\Gamma X$$

   sends the set of $U$-plots of $X$ into the set of set-maps ${\mathrm{Hom}}_{\mathrm{Sh}(\mathrm{CartSp})}(U,X)\mapsto {\mathrm{Hom}}_{\mathrm{Set}}(\Gamma (U),\Gamma (X))$.

   This is noteworthy, because the image-factorization of this map is concretization the map that sends a sheaf to its underlying concrete sheaf: the underlying diffeologial space.

   So we are wondering if this points to a nice general abstract way of understanding how the quasi-topos of concrete sheaves sits inside the topos of all sheaves. Is there any good way to speak for a local topos of the image factorization of the unit map of the extra adjunction?

   And generally: is there known for quasi-toposes any characterization akin to the characterization of sheaf toposes as accessible exact reflective subcategories of presheaf toposes?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 12th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexFor any topos $E$ and any [[Lawvere-Tierney topology]] $j$ on $E$, the full subcategory of $j$-sheaves is left-exact reflective and a topos, while the full subcategory of $j$-separated objects is reflective and a quasitopos. Moreover, the reflection into $sep_j(E)$ can be obtained as the image factorization of the reflection into $sh_j(E)$. In the case in question, the adjunction $\Gamma \dashv Codisc$ forms a geometric morphism $Set \to Sh(CartSp)$, which is in fact a [[geometric embedding]] since $Codisc$ is fully faithful. Therefore, under the equivalence between subtoposes and Lawvere-Tierney topologies, it corresponds to such a topology on $Sh(CartSp)$, and your argument shows that the concrete sheaves are the separated objects for this topology. There are other interesting examples of this sort of situation. For instance, the effective topos admits a geometric morphism $\Gamma\dashv\nabla$ from $Set$ (though in this case $\Gamma$ does not have a left adjoint) whose right adjoint part produces "codiscrete" objects, and the separated objects for the resulting topology are of importance. So the general situation should be described somewhere, but I'm not sure where. > is there known for quasi-toposes any characterization akin to the characterization of sheaf toposes as accessible exact reflective subcategories of presheaf toposes? Well, C2.2.13 in the Elephant implies that any Grothendieck quasitopos (= locally small, cocomplete quasitopos with a strong-generating set) is the category of separated objects for some Lawvere-Tierney topology on a presheaf topos. I don't know a direct characterization of these in terms of properties of the reflector---it preserves finite products and monomorphisms (though not all limits), but I don't know whether that's a characterization.

   For any topos $E$ and any Lawvere-Tierney topology $j$ on $E$, the full subcategory of $j$-sheaves is left-exact reflective and a topos, while the full subcategory of $j$-separated objects is reflective and a quasitopos. Moreover, the reflection into ${\mathrm{sep}}_j(E)$ can be obtained as the image factorization of the reflection into ${\mathrm{sh}}_j(E)$.

   In the case in question, the adjunction $\Gamma ⊣\mathrm{Codisc}$ forms a geometric morphism $\mathrm{Set}\to \mathrm{Sh}(\mathrm{CartSp})$, which is in fact a geometric embedding since $\mathrm{Codisc}$ is fully faithful. Therefore, under the equivalence between subtoposes and Lawvere-Tierney topologies, it corresponds to such a topology on $\mathrm{Sh}(\mathrm{CartSp})$, and your argument shows that the concrete sheaves are the separated objects for this topology.

   There are other interesting examples of this sort of situation. For instance, the effective topos admits a geometric morphism $\Gamma ⊣\nabla$ from $\mathrm{Set}$ (though in this case $\Gamma$ does not have a left adjoint) whose right adjoint part produces “codiscrete” objects, and the separated objects for the resulting topology are of importance. So the general situation should be described somewhere, but I’m not sure where.

   is there known for quasi-toposes any characterization akin to the characterization of sheaf toposes as accessible exact reflective subcategories of presheaf toposes?

   Well, C2.2.13 in the Elephant implies that any Grothendieck quasitopos (= locally small, cocomplete quasitopos with a strong-generating set) is the category of separated objects for some Lawvere-Tierney topology on a presheaf topos. I don’t know a direct characterization of these in terms of properties of the reflector—it preserves finite products and monomorphisms (though not all limits), but I don’t know whether that’s a characterization.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 12th 2010
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   Author: Urs
   Format: MarkdownItexHey, that's great. Thanks Mike! That makes me quite happy.

   Hey, that’s great. Thanks Mike! That makes me quite happy.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 13th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI included some of this now at [[cohesive topos]]

   I included some of this now at cohesive topos

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI added a bit of this to [[concrete sheaf]].

   I added a bit of this to concrete sheaf.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat is "$Ctr$" meant to stand for?

   What is “$\mathrm{Ctr}$” meant to stand for?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexMike, I tried to go through [[concrete sheaf]] and beautify it further. I moved part of what you added to the Definitions-section, where I now keep two definitions: first the elementary one over a concrete site, then then more abstract one in any local topos. I made part of the discussion that you added into a proposition which shows that the more general definition does reduce to the more elementary definition. In the course of this I expanded a bit and ended up re-expressing some of the material you had earlier, to make it fit the new entry design. You should please have a critical look at what I did and see if you can live with this. The remarks about how it follows that concrete sheaves form a quasitopos I kept in the Properties-section in a formal Proposition-environment that replaces the material which was there previously.

   Mike,

   I tried to go through concrete sheaf and beautify it further. I moved part of what you added to the Definitions-section, where I now keep two definitions: first the elementary one over a concrete site, then then more abstract one in any local topos.

   I made part of the discussion that you added into a proposition which shows that the more general definition does reduce to the more elementary definition. In the course of this I expanded a bit and ended up re-expressing some of the material you had earlier, to make it fit the new entry design. You should please have a critical look at what I did and see if you can live with this.

   The remarks about how it follows that concrete sheaves form a quasitopos I kept in the Properties-section in a formal Proposition-environment that replaces the material which was there previously.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex"Ctr" was for the "center" of a local geometric morphism, which is what Johnstone calls the geometric morphism $\Gamma \dashv Codisc$. It's probably not a particularly great name. Your improvements look good; I was just too lazy to try to integrate what I wrote with the rest of the article. (-: I added a few remarks to [[separated presheaf]] mentioning the generalization to arbitrary toposes.

   “Ctr” was for the “center” of a local geometric morphism, which is what Johnstone calls the geometric morphism $\Gamma ⊣\mathrm{Codisc}$. It’s probably not a particularly great name.

   Your improvements look good; I was just too lazy to try to integrate what I wrote with the rest of the article. (-: I added a few remarks to separated presheaf mentioning the generalization to arbitrary toposes.