36,397 reputation
239104
bio website
location
age
visits member for 3 years, 7 months
seen 1 min ago

Jul
13
comment The inverse image of a sheaf
If $j$ is an open immersion things simplify a lot.
Jul
12
comment Isomorphism of presheaves
Yes, that's correct.
Jul
10
comment Algebraic set in affine space
Show that the only compact algebraic subsets of complex affine space are discrete.
Jul
9
comment ind-completion and ordinals
$\mathbf{Ind}(\mathcal{C})$ is basically the category of "formal direct limits" of directed systems in $\mathcal{C}$. Alex's question and my question are closely related, but I am not completely sure they are equivalent. The fact of the matter is, if you have direct limits for all ordinal-indexed directed systems, then you have direct limits for all directed systems – but you have to build them up iteratively!
Jul
9
comment Morphisms of adjunctions in 2-categories
Use the 2-categorical Yoneda embedding to reduce to the case of $\mathfrak{Cat}$.
Jul
9
comment ind-completion and ordinals
More concretely, does every directed poset admit a cofinal map from an ordinal?
Jul
9
comment Fibrewise product
Do you understand the definition given?
Jul
8
comment Proving that finite direct and inverse limits exist in an additive category having kernels and cokernels
Of course, one still has to prove that kernels (in the additive sense) can be used to construct equalisers...
Jul
8
comment Why does tensoring a projective resolution with a flat module give another projective resolution?
@Nishant $P \otimes_R S$ is a projective $S$-module if $P$ is a projective $R$-module.
Jul
8
comment Does something that is injective, surjective or bijective imply that it is a function?
A partial injection is not (always) a function, but I suppose that's more of an example of the red herring principle.
Jul
8
comment Proof that $~\hom_G(V,\bigoplus U)=\bigoplus\hom_G(V,U)$
You mean limit.
Jul
8
comment Being smooth homotopic relation: proof
Hint: Use a bump function...
Jul
8
comment free module implies surjective map of affine schemes
Well, $\{ 0 \}$ is free...
Jul
7
comment Distinguishing sets according to more fine-grained notions than cardinality.
You may like the theory of realisability toposes, especially the part that deals with so-called "modest sets".
Jul
7
comment Adjunction between cocomplete categories
The standard terminology, I believe, is "left biadjoint". I guess the two hom-categories mentioned are also equivalent to $\mathrm{hom}(C^\mathrm{op}, \mathrm{hom}_c (D, E))$, in which case what you will have shown is that the 2-category of cocomplete categories is bitensored and bicotensored. Perhaps you might find some analogous results in the literature about derivators.
Jul
7
comment Help defining the $\mathrm{supp}$ function on free algebraic structures.
Unfortunately the map is not natural in the technical sense. It might be lax or oplax natural, however.
Jul
6
comment Is the continuation monad terminal?
The terminal monad is actually, funnily enough, the constant $1$.
Jul
5
comment Existence of a coproduct and representability of a functor
Your claim is the correct version.
Jul
2
comment What makes “the topos $\mathbf{M}_2$” such a good counterexample?
@CameronWilliams The question is vague and the answers will be a matter of opinion. I vote to close.
Jul
2
comment What makes “the topos $\mathbf{M}_2$” such a good counterexample?
On the contrary, this topos has some rather unusual properties and is, I think, very far from being generic.