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Jan
18
answered Universal property of functor category
Jan
18
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Each rational number is computable, but it is not at all obvious to me that a Cauchy sequence of rational numbers is computable.
Jan
17
comment Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
But what if I only demand that computable Cauchy sequences converge? Even with the Dedekind cut construction, there is a subtlety about what kinds of subsets "exist"...
Jan
17
answered Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology
Jan
17
comment Ring of formal power series finitely generated as algebra?
I don't understand your claim that finitely-generated local $k$-algebras are artinian, but it's certainly true that a local Jacobson domain must be a field. (Because then the unique maximal ideal = Jacobson radical = nilradical = 0.)
Jan
16
answered Surjectivity of projective maps from an inverse limit to an element of the direct product
Jan
16
comment Wedge Product Vocabulary
The notion of support of a differential form $\omega$ already exists. Unfortunately it means the closure of the set of points on which $\omega$ is non-zero. (Recall, a differential form is a smooth assignment of alternating multilinear forms to each point of a manifold...)
Jan
16
comment What is your definition for neighborhood in topology?
There are advantages to both definitions. For example one might like to say that a space is locally compact if for every neighbourhood of a point contains a compact neighbourhood... but compact subsets are usually not open!
Jan
14
comment If an abelian category has a generator then it is well-powered
One definition of Grothendieck category asks that there be a generator. Since Grothendieck categories are locally small and cocomplete, by the remarks here, any generator is automatically a strong generator.
Jan
14
answered If an abelian category has a generator then it is well-powered
Jan
14
answered Eigenvalues and equalizers
Jan
14
comment Direct limits and germs of continuous functions
And, for more or less the same reasons, germs of real analytic functions are classified by the power series. At any rate, the claim that there are continuum-many different germs is easy to show: there are continuum-many different constants, and there are only continuum-many continuous functions that are defined on an open interval around $0$, and there are only countably many open intervals we need to think about because we have a countable base of open sets.
Jan
14
comment Images and preimages over a superstructure.
I agree that taking the naïve union may cause problems. Probably, taking $X$ to be a set of urelements will make things better-behaved... for example, you will be able to determine which level any non-empty set is in by inspecting how many layers of $\{ \cdots \}$ you need to peel off before getting to an atom.
Jan
14
comment Initial structures in the category of algebraic systems of the same type
I don't think that's quite what is meant in the book cited. The "initial structure" being discussed is much closer to the notion of initial lifts in e.g. a topological concrete category.
Jan
14
comment Proving every subset of a finite set is finite
What's your definition of finite set? What's an infinite set? The proof will depend on the details.
Jan
14
answered Examples of Monads and their Algebras
Jan
13
comment What is the free category on the underlying graph of a category?
There is a canonical map from $D$ to the free category on $D$, but it is in general not an isomorphism or an equivalence.
Jan
13
comment What is the free category on the underlying graph of a category?
Suppose given a pair of morphisms $f, g$ with composite $h=g\circ f$. Then in the free category has morphisms $f,g,h$ as well, but now $h\ne g\circ f$.
Jan
13
accepted Cartesian closed categories with double negation elimination
Jan
13
comment What is the free category on the underlying graph of a category?
Almost always. For example, the free category on the empty graph is the empty category; but as soon as there are two composable arrows then the free category is different.