Search Results

Here is what I think is the simplest strategy. I'm only giving a sequence of lemma which lead to the result and I think they are all easy enough, but maybe a little teddious to write down (but let me …

This will not be the case in general. A family is a cover in the canonical topology if all its pullbacks are jointly regular epimorphism. So this will for example be the case if coproducts are univers …

This is essentially correct, and there is no need for the topology to be subcanonical. But let me clarify: Whether the topology is subcaninical or not, we have the following: given any family of maps …

More generally, given a fully faithful functor $i: C \hookrightarrow D$, there is a Grothendendieck topology on $D$ such that the category of sheaves identifies with $Psh(C)$. That topology is given b …

To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)Kock and Moer …

The question is not precise enough: it depends which topology you chose on the category of topological spaces. You will get the same category of sheaves if you are in a situation where Grothendieck's …

That exactness conditions can be rephrased more explicitely as: $$ Hom(V,Z) = \left\lbrace (v_i) \in \prod_i Hom(U_i,Z) \ \middle| \ \forall i,j,v_i \circ \pi_1 = v_j \circ \pi_2 \right\rbrace $$ wher …

Given $H$ a presentable category and $S$ a set of maps in $H$ then the fullcategory $H^S$ of objects in $H$ that are right orthogonal to every arrow in $S$ is a reflective subcategory of $H$. Moreove …

What people usually call a base of the topology is a family $P$ such that if you have a finite set $U_i \in P$ then there is a covering of $\cap U_i$ by elements of $P$. you do not necessarily need $P …

This has nothing to do with $\infty$-categories, but with the fact that we look at the full topos and not an arbitrary site of definition: Theorem: If $T$ is a (Grothendieck) ($1$-)topos, then a "she …

One small additional remark to Qiaochu Yuan response and David Roberts comment to show that it is really the existence of an adjoint that is the important point here. (and that was really too long for …

A small disclaimer: My knowledge of algebraic geometry is relatively basic so I will not discuss anything related to scheme directly. But it seems that most of what you are asking has very little to d …

I'm looking for a reference for the following result: Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over …

It is a slighty tricky question and there is a lot to say, so let's go point by point: 1) As I said in the comment, if you want $F$ to be a subobject of $\Omega$ you need $F(X)$ to identify to a set …

Let $ \chi : X \rightarrow \Omega$ be the characteristic function of $U$. By definition of a subobject classifier, the characteristic function of the pullback of $U$ by $U_i \rightarrow X$ is just th …