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Oct
29
revised Free Metric Space?
deleted 54 characters in body
Oct
29
answered Free Metric Space?
Oct
29
comment ENS is an abbreviation of?…
Incidentally, it also stands for École Normale Supérieure...
Oct
29
comment If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
@PtF It holds if there is an automorphism of $G$ that carries $H_1$ to $H_2$.
Oct
28
answered Visualizing a homotopy pull back
Oct
28
comment When does $\hom(A,-)$ have trivial kernel?
If $A$ is projective and satisfies your condition then it is a strong generator (in the sense that $\mathrm{Hom}(A, -)$ is conservative / faithfully exact): cf this question.
Oct
28
comment On consistency of axiomatic systems
I am using here the convention that free variables have implicit universal quantification. It is certainly the case that we can derive $\lnot (\forall n . n = 0)$.
Oct
27
comment The empty function and constants
That's right. There's a unique function $\emptyset \to X$. But a nullary function is a function $X^0 \to X$.
Oct
27
comment Exactness of functors as “iff”; conjecture about bifunctors
Your new conjecture is still obviously false: the constant zero functor has both a left adjoint and a right adjoint.
Oct
27
comment Are concrete categories whose arrows don't preserve all the structure ever interesting?
One could also make a joke about "forgetting structure" = "equipping costructure"...
Oct
27
answered Are concrete categories whose arrows don't preserve all the structure ever interesting?
Oct
27
comment free object isomorphisms
Hint: consider the abelianisations of $F_1$ and $F_2$ and use the fact that free abelian groups have a well-defined rank.
Oct
27
revised free object isomorphisms
deleted 3 characters in body; edited tags
Oct
27
comment Are concrete categories whose arrows don't preserve all the structure ever interesting?
It does happen sometimes – by accident – because of a phenomenon known as "property-like structure". For instance, it is a fact that every complete meet semilattice is also a complete join semilattice (hence, a complete lattice) in a unique way, so these three classes of structures have the same objects as subcategories of $\mathbf{Poset}$, yet have different homomorphisms.
Oct
27
answered Epi-Mono factorization in presentable categories
Oct
27
comment Exactness of functors as “iff”; conjecture about bifunctors
Your conjecture about bifunctors is obviously false: take the functor that sends everything to zero. It is exact but fails to have any interesting reflection properties.
Oct
27
comment If a functor preserves finite limits, does it preserve subobjects?
I don't feel the definition of subobject as an equivalence class is very useful.
Oct
27
answered If a functor preserves finite limits, does it preserve subobjects?
Oct
27
comment An 'easy' way to prove that epimorphism of sheaves implies surjectivity on stalks
That is not at all what I am saying. I am saying, knowing that the stalks are jointly conservative and that they are exact implies that they jointly reflect exactness.
Oct
27
comment Exactness of functors as “iff”; conjecture about bifunctors
It is true that $\mathrm{Hom}(A, -)$ is left exact for all $A$ and it is true that $\mathrm{Hom}(A, \mathcal{S})$ being exact for all $A$ implies $\mathcal{S}$ is split exact. But these two facts are unrelated.