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Aug
12
comment Are pulation squares “weak equivalences”?
The only categories I know with this property are the empty category and the trivial category $\mathbb{1}$. $\mathbf{Set}$ doesn't have this property, $\mathbf{Ab}$ doesn't have this property, most lattices don't have this property, etc.
Aug
12
comment Are pulation squares “weak equivalences”?
Is it obvious that it has the 2-out-of-3 property? Pullbacks and pushouts each have (a different) half of the 2-out-of-3 property...
Aug
12
comment Localisation of a binary product of categories
It does, but it's a bit complicated. The linked discussion at the nForum has more details: it's some manipulation of adjoints plus facts about reflectors of exponential ideals.
Aug
12
comment How are sets “trees with no symmetries”
Well, a set as conceived in ZF is a thing which has other sets as members. So one could draw the membership graph as a tree.
Aug
11
comment Localisation of a binary product of categories
Didn't you get an answer to this question at MO already?
Aug
11
comment Monomorphisms and epimorphisms in the category of Boolean algebras
Monomorphisms are always injective, because the forgetful functor $\mathcal{B} \to \mathbf{Set}$ has a left adjoint.
Aug
10
comment “Coordinate functions” on the structure-sheaf definition of a smooth manifold
possible duplicate of Functionally structured spaces and manifolds
Aug
9
comment How does indexing work in EGA/ how to search for a result in EGA?
Indeed, the cited result is in the 1971 edition.
Aug
9
comment Can we define 'model' to mean 'diagram' in the sense of mathematical logic? If so, how to define the satisfaction relation?
Yes, but there is no guarantee that every element in a model of the diagram is the interpretation of a constant!
Aug
8
answered How to define the category of model structures of a category?
Aug
8
revised How to define the category of model structures of a category?
edited tags
Aug
8
comment Functionally structured spaces and manifolds
Well, to be more precise, it has to be a subsheaf considered as a sheaf of $\mathbb{R}$-algebras, and as you say it has to be non-trivial if the space is non-empty.
Aug
8
comment Functionally structured spaces and manifolds
What you call a "functional structure" is more commonly known as a "subsheaf of the sheaf of continuous real-valued functions". Your question is local and can be phrased without recourse to such complications, though. I think you should probably add some other hypotheses: for example, if $X$ is a one-point space then it automatically satisfies your condition. Perhaps you should ask that the smooth function $g$ be unique.
Aug
8
comment Grothendieck topology of sets and Cech cohomology
My point is that you can't even get the correct answer in degree $0$: $\check{H}^0 (X, \mathcal{A})$ is always isomorphic to $H^0 (X, \mathcal{A})$, which is just the set of $\Gamma$-equivariant maps from $X$ to $A$.
Aug
8
comment Grothendieck topology of sets and Cech cohomology
For this to work you need $X$ to be the singleton set: consider the case where $\Gamma$ is the trivial group!
Aug
8
answered What is a (-1)-morphism?
Aug
8
revised What is a (-1)-morphism?
edited tags
Aug
6
comment Vector space bases without axiom of choice
Just because we can't describe the basis doesn't mean it doesn't exist! For this particular example, if the axiom of choice holds up to a sufficiently large cardinal, then $F^{\mathbb{N}}$ would have a basis. But the axiom of choice could still fail higher up.
Aug
6
comment Objects with a “Homogeneity Principle”
That looks more like the property of being "Galois"!
Aug
6
comment What are the main differences between set theory versus pure type systems?
The Wikipedia article is short on details. Have you looked at the nLab article?