Zhen Lin
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 8h comment How to write the real projective plane as a pushout of a disk and the mobius strip? The Möbius strip is obtained by gluing together two antiparallel edges of a square. You can express this as a pushout fairly straightforwardly. 2d comment Intuition for the definition of a dense functor? I'm not sure there is actually a connection of the type you are looking for. Apr 29 comment Is an “$\aleph_0$-limit” a finite limit or a small limit? It is conventional to say that a category is $\kappa$-small if it has $< \kappa$ (objects and) morphisms. Apr 28 comment Reference/Definition of Homotopy in an Abstract Category The truncation of the simplicial localisation always recovers ordinary localisation. (This is an instance of the fact that the left adjoint of the composite is the composite of the left adjoints.) Apr 28 comment How to call a “non-strict” monoidal category? @NajibIdrissi In fact, such a concept exists, but for some reason it is called a skew monoidal category. Apr 25 comment Differentiable manifolds as locally ringed spaces Perhaps I mean here specifically. Apr 23 comment Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms' I believe it is conventional to take $F$ to be the so-called fundamental fibration, i.e. $X \mapsto \mathcal{C}_{/ X}$. This is a 2-presheaf. Apr 22 comment if $F_{\bullet}$ is a complex and $r$ an integer, what is $F_{r-\bullet}$? For me, if $F_\bullet$ is a chain complex then $F_{r-\bullet}$ is a cochain complex where $(F_{r-\bullet})_n = F_{r-n}$. Note that the differentials now increase degree. Apr 22 comment Presheaf and copresheaf categories on finite sets $\mathbf{Set}$, $[\mathbf{FinSet}, \mathbf{Set}]$ and $[\mathbf{FinSet}^\mathrm{op}, \mathbf{Set}]$ are all distinct. Apr 20 comment Terminal object in the category of sheaves? The initial object is not the constant presheaf $0$ but rather the sheafification thereof. Apr 19 comment When are sheafification and the embedding of sheaves into presheaves exact functors? Yes, it is exact for groups. More generally, also for any finitary algebraic theory. Apr 19 comment When are sheafification and the embedding of sheaves into presheaves exact functors? Very rarely. That would make cohomology trivial. Apr 19 comment When are sheafification and the embedding of sheaves into presheaves exact functors? Sheafification is exact for sheaves of abelian groups, hence also for sheaves of $R$-modules or $\mathscr{O}_X$-modules. Apr 17 comment Commutations of pullbacks and coproducts? Why go to the trouble of bringing up cartesian closedness or convenient categories? The isomorphisms in question all basically arise from the fact that coproducts are disjoint unions. Apr 15 comment On the existence of finite tensors/cotensors Tensors (resp. cotensors) in ordinary categories are special cases of coproducts (resp. products) Apr 14 comment Is the category of categories a topos? It is not a 2-topos in the sense of Lurie (which is a (2, 1)-category). Apr 14 comment Why isn't '&' used for logical conjunction? Linear logic also has ⅋, to make things more fun. Apr 13 comment Extra hypotheses in proposition in Sheaves in Geometry and Logic? 1. Yes, completeness is superfluous – you just need finite limits. 2. One must be clear about which pullback functor has a right adjoint. If $\mathcal{C}_{/ B}$ is cartesian closed then $B' \times_B (-) : \mathcal{C}_{/ B} \to \mathcal{C}_{/ B}$ has a right adjoint, but that's not the claim. 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.