Zhen Lin
Reputation
93/100 score
 May5 answered Which axiom of set theory does this formula represent ? Why? May5 revised Which axiom of set theory does this formula represent ? Why? edited tags May5 answered Are sections of vector bundles $E\to X$ over schemes necessarily closed embeddings? May5 comment what is the precise definition of a morphism defined over $k$? Well, instead of thinking about automorphisms, first generalise to morphisms between arbitrary varieties over $k$. Then specialise to affine varieties and use the correspondence with $k$-algebras. May5 comment Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings? @yogeshmore No, it is not. A morphism of affine schemes is étale if and only if the corresponding ring homomorphism is étale. May5 answered Is the skeleton-coskeleton adjunction $sSet$-enriched? May4 comment Covariant and Contravariant Functor of Fixed Set Question - Category of Sets The question is imprecise. After all, the formulae $X \mapsto X^A$ and $X \mapsto A^X$ do not specify what happens to morphisms. The point is to define actual functors whose object parts are as specified – so there's really nothing much to it at all. May2 answered when a presheaf is a sheaf May2 comment Classifying space infinite totally ordered set contractible @NajibIdrissi $\omega_1$ has a bottom element, so its nerve is contractible (in the strong sense). More generally, any category with an initial object has contractible nerve. May1 accepted Positive and negative logical connectives May1 comment Adjunction between topological and simplicial presheaf categories $\mathcal{C}^\Delta$ is not good notation, so I will write $\mathcal{D}$ instead. It is not hard to see that $\mathbf{Top} \to \mathbf{sSet}$ induces a functor sending topologically enriched functors $\mathcal{C} \to \mathbf{Top}$ to simplicially enriched functors $\mathcal{D} \to \mathbf{sSet}$, but I don't see any candidate for a left adjoint. (Actually, the only functors going the other way that I can think of are either constants or factor through the homotopy category.) May1 comment Tor Functor Commutes with Direct Limits Classically, "direct limit" means "colimit of a directed diagram". In particular, they are filtered colimits. Apr30 comment Can I define a predicate over a set and use it in the definition of another one? Are you trying to work in a specific axiomatic set theory? Apr30 comment How to understand cocategories For any $\mathcal{C}$ with pullbacks, cocategories in $\mathcal{C}$ are the same as functors $\mathcal{C} \to \mathbf{Cat}$ whose composites with $\operatorname{ob}, \operatorname{mor} : \mathbf{Cat} \to \mathbf{Set}$ are representable. Such functors are not so common, but there are a few examples, e.g. the left and right adjoints of $\operatorname{ob} : \mathbf{Cat} \to \mathbf{Set}$. Apr30 comment Characterization of epic morphisms in the category of rings. See this question. Apr30 comment Points of scheme with residue field $k$ vs $k$-point That's not possible when all the morphisms are $k$-morphisms. Apr29 comment Choice of a skeleton Actually, in my mind, a skeleton is not just a full subcategory but also equipped with a quasi-inverse to the inclusion. At any rate, all you need is a sufficiently strong axiom of choice (and the law of excluded middle). Apr29 comment Is $S^1 \times S^1$ really a torus? The torus has many (Riemannian) metrics. One of them is even flat! Apr29 revised Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms? deleted 15 characters in body Apr29 revised $\{P\}^-$ notation in Algebraic Geometry edited tags