Search Results

Projective limits in vector spaces and in sets are the same so the stalk does not depend on whether you consider this as a co-presheaf of sets or vector spaces. in both case it is just the directed p …

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 …

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 …

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 …

The only important axiom in order to define a notion of sheaf is the stability under pullback. There is a proposition in SGA4 saying that if you have a family of sieves only satisfying the pullback st …

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 …

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 …

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 …

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 …

The problem is that your definition is well behaved only if there is enough open subsets $V$ such that $V$ is connected (if there is no such open subset, then your condition is empty) hence the notion …

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 …

The notion of "Cauchy real" is always a bit ambiguous: it depends on what you call a Cauchy sequence. For the argument that follow I need a notion of Cauchy sequence that is geometric (is classified b …

So let $G$ be a topological group, $X$ a topological space, and let $\mathcal{G}$ be the sheaf of local functions from $X$ to $G$ (which is a sheaf of group over $X$). Let $T$ be a (locally trivial) …

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 …

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 …