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You already proved that: $$ Hom_C(X,Y) \simeq (ay Y)( X)$$ But $Hom_C(X,Y)$ is by definition $(y Y)(X)$, and all those isomorphism are functorial in $X$ so you have proved that $(y Y)( X) \simeq ay …

$(i) \Leftrightarrow (ii)$ is true and is Proposition C.2.4.14 in Peter Johnstone's Sketches of an elephant. More generally he shows that a bounded geometric morphism $f: \mathcal{E} \to \mathcal{S}$ …

It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exist, as any Grothendieck topos is the classifying topos of something …

I think your formulation of the ultrafilter principle implies the de Morgan law. Let $U$ be any proposition, consider the boolean algebra $A$ freely generated by an element $v$. so $A = \{ 0,1,v,\ne …

Hello ! If I'm not mistaken, an atomic topos decompose as a disjoint sum of connected atomic topos, and Connected Atomic topos with a point corresponds to classifying topos of localic groups. But wh …

Let $A$ be a small allegory (like in Freyd and Scedrov book, or in the Elephant of Johnstone), does it always exists a tabular allegory $B$ and a fully faithfull representation of $A$ in $B$ ? I am e …

Ok I think I finally found an internaly valid proof by my self, so I explain it briefly here in case someone is interested some day : If $U \in \Omega$ is a subterminal object, you can define the ele …

$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$. $i_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, t …

There are two observation to be made that limit a little this kind of analogy: 1) Site are indeed in some sense presentations, but an infinity theory (I mean with operation of infinite arity) somethi …

So the final answer, is 'no', but there is still something interesting to say: Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal …

There is a more general form of Day's theorem that does pretty much that, at least for sub-canonical topologies: Theorem (Day): Let $C$ be a complete and co-complete Category, and $D \subset C$ a ful …

Here are some examples, that should show you that there is a lot of countable toposes and lot of things in between finite sets and sets, too much to actually hope to list or classifies. First as I s …

Hi ! I'm considering in a general topos $T$ the object $R$ of lower semi-continuous real (one sided lower non-empty Dedekind cuts, as for exemple in http://ncatlab.org/nlab/show/one-sided+real+number …

Here are some examples : For any topological space $X$, the topos of sheaf $\operatorname{Sh}(X)$ has enough points. In most cases this is not a coherent topos. If I remember correctly (for $X$ sober …

Ivan's example in the comment actually proves that all the questions have negative answers. As observed by Ivan, in the category of pointed set, there is a subobject classifier given by $\{*\} \to \{* …