Zhen Lin
Reputation
43,201
392/400 score
 Apr25 comment Are complex submanifolds necessarily closed? It is conceivable that ‘submanifold’ means closed submanifold. I attended a lecture course on complex manifolds using that definition. Apr25 answered Theory of promonads Apr25 comment Theory of promonads Hmmm, I'm not convinced your characterisation is correct. A promonad on a small category $\mathbb{D}$ should correspond to a monad on $[\mathbb{D}^\textrm{op}, \textbf{Set}]$ whose underlying endofunctor preserves all colimits. Apr25 comment Is there a category whose objects are sets, in which both products and coproducts are Cartesian products in the classical sense? Easy: let $\mathcal{C}$ be the category whose objects "are" sets, and whose morphisms $X \to Y$ are the group homomorphisms from the free abelian group generated by $X$ to the free abelian group generated by $Y$. This has the required property. Apr24 answered If monic, then *property*. Does the converse hold? Apr24 comment pullback square of regular epimorphisms is a pushout It's just the statement that if $A = B \cup C$, then $A$ is the pushout of $B \leftarrow B \cap C \rightarrow C$ as well. Apr24 comment What is the homotopy colimit of the Cech nerve as a bi-simplical set? The diagonal of a bisimplicial set is weakly equivalent to its homotopy colimit: see here. Apr23 comment Can we think of an adjunction as a homotopy equivalence of categories? Homotopy, defined in the classical sense for topological spaces using the standard interval, is an equivalence relation, and it is clear that is what the OP is thinking about. (Also note that in the model structure on $\textbf{Cat}$, two functors are homotopic precisely if they are isomorphic, not merely linked by a natural transformation.) Apr23 comment Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics? @HaraldHanche-Olsen It's not entirely useless. It can be used to prove a whole range of negative results, like the non-existence of large cardinals! Apr22 comment Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces? I do not know of an example off the top of my head. CW complexes are much nicer than general topological spaces, so it is more plausible that the claim is true in that case. Apr22 answered Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces? Apr22 comment Homotopical equivalence of complexes You have defined the notion of quasi-isomorphism. Every chain homotopy equivalence is a quasi-isomorphism but not vice versa. Apr22 comment How does one charaterize functionhood (etc.) in the category of relations? $\textbf{Poset}$ is a subcategory of $\textbf{Cat}$ that is closed under finite products, so a category enriched over $\textbf{Poset}$ is also enriched over $\textbf{Cat}$. Apr22 answered How does one charaterize functionhood (etc.) in the category of relations? Apr22 answered pullback square of regular epimorphisms is a pushout Apr22 answered How Co-Comma Categories are constructed? Apr22 revised How Co-Comma Categories are constructed? added 2 characters in body Apr21 comment pullback square of regular epimorphisms is a pushout I've not seen any lemma like that, but there is one involving a pullback–pushout square of monomorphisms. Apr21 comment French translation of “well-powered” category @Pece In SGA 4 they seem to just say ‘l'ensemble des sous-objets de $X$ dans $C$ est petit’... Apr21 comment French translation of “well-powered” category @user42912 No, that refers to "full subcategories".