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Jul
21
comment Two definitions of functions
The second notion, modulo the evident notion of equivalence, is properly called a function class.
Jul
20
comment What kind of points are there in a finite type $k$-scheme?
The generic point is usually not open. Think of an algebraic curve, for example.
Jul
19
comment in which conditions the following holds: If the pullback of a morphism is an isomorphism, then this morphism is an isomorphism?
If your category is a regular category, then it suffices that $p$ be a regular epimorphism. More generally, if you have pullbacks, then it suffices that $p$ be a descent epimorphism.
Jul
18
comment Projective roots of a homogeneous polynomial
$(0 : 1)$ is a zero if and only if $F_n (Y) = 0$; but in that case, $F (X, Y)$ is divisible by $Y$, etc.
Jul
18
comment Direct limits and pullbacks
Only in suitable categories!
Jul
18
comment Model existence for infinitary logics
Please feel free!
Jul
17
comment Is it possible to find square root using only rational numbers and elementary arithmetic operators
As I said, it just means that the sum, difference, product, or quotient of two rational numbers is again a rational number. I'm sure you can find a proof of this yourself.
Jul
17
comment Proper terminology for a pair of magmas
This is just a set with two binary operations.
Jul
17
comment Examples and definition of cocompact objects
I believe the only cocompact objects in $\mathbf{Set}$ are $0$ and $1$.
Jul
17
comment Examples and definition of cocompact objects
Cocompactness is not as useful as compactness because many categories of interest are $\mathbf{Ind}(\mathcal{A})$ for some $\mathcal{A}$, but not so many are $\mathbf{Pro}(\mathcal{A})$.
Jul
16
comment What makes a Lie Group a Differentiable Manifold?
It's part of the definition!
Jul
15
comment Algebras of the environment monad
@QiaochuYuan I infer from the description that the environment monad is the monad induced by the unique comonoid structure on $E$ (wrt the cartesian monoidal structure).
Jul
15
comment Homotopy limits
Actually, one should first replace $f : X \to Y$ with a fibration between fibrant objects, not just a morphism between fibrant objects.
Jul
15
comment Homotopy limits
The point is that a cone in the triangulated category corresponds to a homotopy kernel in the model category. Thus the problem boils down to verifying that the well-known product-kernel formula for limits is valid in this case. See also this question on MO.
Jul
14
comment Could induction give us an infinite sequence of sets $X_1 \subseteq X_2 \subseteq \cdots$ or do we need the axiom of choice?
In the proof of König's lemma, at each step, you have many candidates for $X_n$, and it matters which one you "choose". Moreover the set of candidates for $X_{n+1}$ will depend on your "choice" of $X_n$.
Jul
14
comment Can the complement of a subset be realized as a limit or colimit?
Complements are neither limits nor colimits, but as an exercise you might like to verify that any map between boolean algebras that preserves joins and meets must also preserve complements.
Jul
14
comment Describing the Wreath product categorically.
The only places (with one exception) where I have seen weak-by-default are where $n$-categories are considered for $n > 2$. You are free to ignore the established literature on 2-category theory if you like, but that would be a pity.
Jul
14
comment Describing the Wreath product categorically.
@QiaochuYuan As Mike Shulman likes to point out, if you think of groups as pointed connected groupoids then there is no interesting 2-categorical structure; but in classical logic, we are free to think of groups as connected groupoids, which is where the 2-categorical structure you are using comes from. Also, conventionally, 2-colimits are "strict" (i.e. involve an isomorphism of categories), so what you are referring to is really a bicolimit.
Jul
14
comment Is it true that $\omega=\{0,1,2,3,\ldots\}$ in ZFC?
The answer to your last question is a straightforward yes: use the compactness theorem.
Jul
14
comment Representation theorem for Heyting algebras?
The difficulty, in some sense, is the Heyting implication. Every Heyting algebra is a distributive lattice, and every distributive lattice can be embedded in a complete Heyting algebra, but this is an embedding of distributive lattices, not Heyting algebras. On the other hand, not every distributive lattice can be embedded as a subalgebra of the distributive lattice of open subsets of a topological space – this is the question of whether there are "enough points".