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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2010
    • (edited Oct 12th 2010)

    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 Sh(CartSp)CodiscΓSetSh(CartSp) \stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}} Set,

    where for SS a set, CodiscSCodisc S is the codiscrete smooth structure on SS in that for UCartSpU \in CartSp we have

    Hom Sh(U,CodiscS)Hom Set(Γ(U),S), 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 UU into the set SS.

    Now the observaton is that the unit of this adjunction

    XCodiscΓX X \to Codisc \Gamma X

    sends the set of UU-plots of XX into the set of set-maps Hom Sh(CartSp)(U,X)Hom Set(Γ(U),Γ(X))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?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 12th 2010

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

    In the case in question, the adjunction ΓCodisc\Gamma \dashv Codisc forms a geometric morphism SetSh(CartSp)Set \to Sh(CartSp), which is in fact a geometric embedding since CodiscCodisc is fully faithful. Therefore, under the equivalence between subtoposes and Lawvere-Tierney topologies, it corresponds to such a topology on Sh(CartSp)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 Γ\Gamma\dashv\nabla from SetSet (though in this case Γ\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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2010

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

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2010

    I included some of this now at cohesive topos

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeOct 18th 2010

    I added a bit of this to concrete sheaf.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2010

    What is “CtrCtr” meant to stand for?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2010

    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.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeOct 18th 2010

    “Ctr” was for the “center” of a local geometric morphism, which is what Johnstone calls the geometric morphism ΓCodisc\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.

    • CommentRowNumber9.
    • CommentAuthorZhen Lin
    • CommentTimeJun 15th 2014
    • (edited Jun 15th 2014)

    What is the connection between concrete presheaves and ¬¬\neg \neg-separated presheaves, in the case where the Grothendieck topology on the site is trivial? Famously, for simplicial sets, the concrete objects are precisely the ¬¬\neg \neg-separated presheaves.

    Here are some hypotheses that make the two notions coincide. Let 𝒞\mathcal{C} be a small category.

    1. 𝒞\mathcal{C} has a terminal object 11.
    2. Every object in 𝒞\mathcal{C} has a point.
    3. Any two points of an object in 𝒞\mathcal{C} are either equal or disjoint, in the sense that if x,y:1Ax, y : 1 \to A are morphisms in 𝒞\mathcal{C} and xyx \ne y, then there do not exist u,v:T1u, v : T \to 1 such that xu=yvx \circ u = y \circ v.

    For example, 𝒞\mathcal{C} could be Δ\mathbf{\Delta}, or 𝒞\mathcal{C} could be a small well-pointed elementary topos minus any initial objects. If I’m not mistaken, conditions (1) and (2) ensure that the family of all points of an object in 𝒞\mathcal{C} is a covering family in the ¬¬\neg \neg topology, and condition (3) ensures that the sheaf condition is trivial. Thus, a ¬¬\neg \neg-separated presheaf on 𝒞\mathcal{C} is concrete. On the other hand, ¬¬\neg \neg-covering families can be recognised by checking surjectivity on points, so concrete presheaves coincide with ¬¬\neg \neg-separated presheaves.

    In fact, writing :Set[𝒞 op,Set]\nabla : Set \to [\mathcal{C}^op, Set] for the right adjoint of Γ:[𝒞 op,Set]Set\Gamma : [\mathcal{C}^op, Set] \to Set, the conditions appear to ensure that the local operators ¬¬\neg \neg and Γ\nabla \Gamma coincide. Can we get a complete characterisation of all those 𝒞\mathcal{C} for which this happens?