Zhen Lin
Reputation
51,605
99/100 score
 Apr 13 revised Cartesian closed category isomorphism added 7 characters in body Apr 10 comment A question about colimits in enriched categories There seems to be some confusion here. If you have tensors (resp. cotensors) then your ordinary limits (resp. colimits) are enriched conical limits (resp. colimits); but you still need to have cotensors (resp. tensors) in order to get all weighted limits (resp. colimits). Apr 9 comment Establish canonical isomorphism $Z^{Y \times X} \cong (Z^Y)^X$ for objects $X, Y$ and $Z$ from $\mathcal{AB}$ category of Abelian groups The first isomorphism listed is false. Apr 7 comment Limit-preserving functor that factors through representable functor. No, that won't work. You should use the hom-set definition of limit here. Apr 7 answered how do contravariant 2-functors preserve adjunctions? Apr 7 comment how do contravariant 2-functors preserve adjunctions? It depends on what you mean by "contravariant", because there are two compositions in a 2-category. Apr 7 comment Categories of defintion for sites on spaces and sites on schemes It's a cultural thing. In algebraic geometry one works over a base scheme, which may be something simple like a "point" ($\operatorname{Spec} k$ for some algebraically closed field $k$) or something more complicated like a space of parameters. Apr 4 comment Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$” Q1 is not really answerable for two reasons: first, because the standard notion of internal logic does not handle predicates with object variables – for that you need stack semantics – and second, because stack semantics presupposes that your predicates are local in the sense you are asking about. Apr 4 revised Sheaf cohomology via resolutions vs. derived categories edited tags Apr 4 comment Sheaf cohomology via resolutions vs. derived categories I like to believe that the principle of conservation of work applies even in pure mathematics. The derived category approach is elegant but is difficult to understand concretely; the resolution approach is concrete but difficult to understand elegantly. Apr 4 comment elementary question concerning definition of sifted colimit Why do you bring up coends? You could just as well use the notation $\varinjlim_\mathcal{C} \varinjlim_\mathcal{D}$ and $\varinjlim_{\mathcal{C} \times \mathcal{D}}$. Apr 4 comment Categorical interpretation of equality type The equality type of $A$ is interpreted as the diagonal $A \to A \times A$ (regarded as an object over $A \times A$). Apr 3 comment Flat Modules are Filtered Colimits of Free Modules Between 2 and 3 you pass from $\mathbf{Ab}$-valued profunctors to $\mathbf{Set}$-valued profunctors. Why is this allowed? Apr 1 comment Representable Functors and Upper Sets (Final Segments) This is a wild guess, but it seems to me that you have some misconceptions about categorification. Apr 1 comment Representable Functors and Upper Sets (Final Segments) That's not what the statement is about. Apr 1 answered Representable Functors and Upper Sets (Final Segments) Mar 31 answered Why are left/right proper model categories called so? Mar 31 comment What is the most general category in which exist short exact sequences? While it makes sense to talk about short exact sequences as soon as you have kernels and cokernels, whether or not it is a useful concept is another story... Mar 31 comment Why are left/right proper model categories called so? There's that. More generally, colimits and cofibrations have to do with the left (first) variable of the hom functor while limits and fibrations have to do with the right (second) variable of the hom functor. Mar 31 comment Why are left/right proper model categories called so? It's the same left/right as left/right adjoint (but the opposite of left/right exact, unfortunately).