Search Results

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 …

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 …

Here are two (related) interpretation of this quote I can think of: A first interpretation is just that Grothendieck was attach to the idea that you can study a 'space' (whatever this mean) by studin …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

I am not very familiar with the analytic side of the pictures or with the analytification functor but here is what I can claim, it seems from your comment that this answer your question: If you have t …

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 …

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 …

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 …

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 …