Zhen Lin
Reputation
99/100 score
 Jan 4 comment category of modules a Grothendieck category If and only if $R$ is trivial. Jan 3 comment Projective object in the category of chain complexes That works fine. Try lifting $(a, b) = (0, 1)$ in $C$. Jan 3 comment Reference for Cech cohomology on sites (not pre-topologies) The only reasonable condition for that is to ask that $\check{\mathscr{C}}{}_{\bullet} (U)$ be a projective resolution, and the only reasonable condition for that is to ask that products of representable presheaves be projective – a condition that is automatic if products of representable presheaves are representable! Jan 3 comment I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? It means every hom-set has a commutative monoid structure, and composition is bilinear. Jan 3 revised I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? edited tags Jan 3 answered I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? Jan 3 comment I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? @user18921 You need the fourth axiom. $\mathbf{Grp}$ satisfies the first three axioms but is not enriched in $\mathbf{Ab}$. Jan 3 comment The category of models of a commutative algebraic theory. For the proof I had in mind, yes. But there are other proofs that generalise to the $\kappa$-ary, many-sorted case. Jan 3 answered The category of models of a commutative algebraic theory. Jan 2 comment How should we think of maps to the intial object? There is no "picture". In all geometric examples, initial objects are "empty", in the sense that every morphism to an initial object is an isomorphism. Jan 2 answered Reference for Cech cohomology on sites (not pre-topologies) Dec 31 comment Can an Element of an Algebraic Structure have Multiple Identities? If a given binary operation has a left identity and a right identity, then they are equal. But rings and fields have more than one binary operation. Dec 30 revised The “closed” subspaces of topological algebraic structures added 108 characters in body Dec 29 comment Zorn's lemma in categorical language @WilliamBallinger That's not quite correct. A maximal element in the sense of Zorn's lemma is not necessarily a maximum element. Dec 29 awarded Yearling Dec 27 comment In categorial logic, why do we need finite products to define the notion of “group,” but not “monoid”? That's not entirely true. You could define "groups" within the doctrine of symmetric monoidal categories, but people prefer to call such a thing a Hopf algebra. Dec 24 comment A full embedding in a finitely complete category Yes. Take the smallest full subcategory of $[\mathcal{C}, \mathbf{Set}]^\mathrm{op}$ containing the representables and closed under finite limits. Dec 23 comment Can this be proved purely on base of UMP? Yes; the product projections are jointly monic. Dec 21 comment Trying to understand the fibre product in the category of spaces over $X$ You're looking at the wrong functor. You should be looking at morphisms into the fibre product. Dec 21 revised Existence proof of the tensor product using the Adjoint functor theorem. edited tags