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Feb
17
comment A pedantic question about defining new structures in a path-independent way.
1. You can perfectly well formulate "this group has cardinality 10" in first-order logic. You could even say "this group has cardinality $\aleph_{42}$ if you allow infinite disjunctions in your logic. 2. Your goal is fundamentally impossible. It is the equivalent of asking for a language-independent way of expressing thoughts: there is no such thing by definition of language; any method of communicating ideas is a language. But maybe you are thinking in terms of the instance–implementation–interface distinction in software engineering, in which case what you ask for is [...]
Feb
17
comment In what sense is the forgetful functor $Ab \to Grp$ forgetful?
No, here "algebraic theory" refers specifically to Lawvere theories, which come with an obvious notion of morphism.
Feb
16
comment Bijection between a variety $X$ and $\hom(k[X], k)$
Choose an embedding of $X$ into affine space, or equivalently, choose generators for the coordinate ring. Then consider the image of the coordinate functions.
Feb
16
comment Definition(s) of Stack
What I meant was that $\textbf{Pic}(U)$ and $\textrm{Pic}(U)$ are not isomorphic as groupoids. But if $\textbf{Pic}(U)$ is skeletal, then there is an obvious bijection between the objects of $\textbf{Pic}(U)$ and the elements of $\textrm{Pic}(U)$.
Feb
15
revised adjunction relation
added 12 characters in body
Feb
15
answered adjunction relation
Feb
15
answered Is there a categorical construction of the general linear group?
Feb
15
comment adjunction relation
You could make it into a simplicial topological space if $X_n$ and $Y_n$ are sufficiently nice, but you can definitely make it into a simplicial set, using this end formula: $$\textrm{Hom}(X, Y)_n = \int_{[k] : \mathbf{\Delta}} \textbf{Set}(\mathbf{\Delta}([k], [n]), \textbf{Top}(X_k, Y_k))$$
Feb
15
comment Projective objects in the category of rings
How can $0$ be projective? Any ring homomorphism to $0$ is surjective, but the only split epimorphisms to $0$ are isomorphisms.
Feb
15
comment How we can understand one category is small
There is nothing specific to category theory about this problem. But here are some general guidelines on when a class $A$ is a set in ZF: 1. If $A$ has members of arbitrarily large cardinality, then it is not a set. 2. If $A$ has members of arbitrarily large rank, then it is not a set. etc.
Feb
15
comment A pedantic question about defining new structures in a path-independent way.
It's more or less the same. For every constant $a$ in the first theory, you need a one-place predicate $\phi_a (x)$ in the second theory such that $\phi_a (x)$ holds if and only if $x$ is the "interpretation" of $a$ in the second theory; and more generally for every $n$-ary operation $\omega(x_1, \ldots x_n)$ in the first theory you need a predicate $\phi_{\omega} (y, x_1, \ldots, x_n)$ in the second theory that holds if and only if $y = \omega (x_1, \ldots, x_n)$ holds in the interpretation, etc.
Feb
15
comment A pedantic question about defining new structures in a path-independent way.
I'm not talking about a group in ZFC or whatever set theory. As you say, ZFC sees "too much". I'm talking about the theory of groups as axiomatised directly in first (or higher) order logic. Also, bi-interpretability isn't a relation between models but rather a relation between theories.
Feb
15
comment A pedantic question about defining new structures in a path-independent way.
Your stated goal sounds absurd. You can't say anything precise about some notion if you don't have a precise definition of that notion! As for your new example, the sentence "every group is an ordered triple" is illegitimate: you can't talk about the signature itself within a formal system. Perhaps what you are really interested in is bi-interpretability.
Feb
15
comment Projective objects in the category of rings
$\{ 0 \}$ isn't projective, unless your ring homomorphisms aren't unital. I guess you are asking about projective objects with respect to the class of all epimorphisms; otherwise, if you are happy to talk about projective objects with respect to all regular epimorphisms, then the projective rings are precisely the retracts of freely-generated rings.
Feb
14
comment A pedantic question about defining new structures in a path-independent way.
Your example is misleading. It just so happens that the two axiomatisations of topological spaces you chose have compatible signatures, in the sense that they are both of the form (set, subset of power set). Of course things go pear-shaped if you interpret "open" in the first axiomatisation to mean "closed" in the second axiomatisation. So I say the problem you bring up is spurious.
Feb
14
comment Terminology Question: Precompose vs Compose?
"Precompose with $f$" usually means the operation $(-) \circ f$. "Pre" here refers to the order of application and not the order in which the symbols are printed on paper...
Feb
14
answered Are the right derived functors of the inclusion sheaf cohomology?
Feb
12
answered Definition(s) of Stack
Feb
12
revised What is the minimum required background to understand articles in the nLab?
deleted 1 characters in body
Feb
12
comment Why isn't the union of the coordinate axes a manifold, using the formal definition of a manifold?
What if we rotate the graph?