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Nov
29
comment How to turn the subclass of all epimorphisms of a category, into a category?
Of course, you could also form the non-full subcategory of $\mathcal{C}$ whose morphisms are the epimorphisms. This makes sense because the composite of two epimorphisms is another epimorphism.
Nov
29
comment Kan fibrations and surjectivity
As hinted by Aaron, you can also get counterexamples if $Y$ is not connected, e.g. the coproduct insertion $\Delta^0 \to \Delta^0 \amalg \Delta^0$.
Nov
28
comment What fragment of ZFC do we need to prove Zorn's lemma?
Mathias uses the usual formulation, i.e. the existence of a choice-function for families of non-empty sets. But this is the same as asking for surjections to split in the presence of disjoint unions.
Nov
28
comment What fragment of ZFC do we need to prove Zorn's lemma?
... apparently I can't read. Mathias mentions in his paper about Mac Lane set theory that the well-ordering theorem is provable in RZC minus Foundation and Infinity. Still, an explicit proof would be nice.
Nov
28
revised What fragment of ZFC do we need to prove Zorn's lemma?
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Nov
28
comment What fragment of ZFC do we need to prove Zorn's lemma?
Good point. I've clarified what I mean by the axiom of choice here.
Nov
28
revised What fragment of ZFC do we need to prove Zorn's lemma?
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Nov
28
asked What fragment of ZFC do we need to prove Zorn's lemma?
Nov
28
answered “The purpose of being categorical is to make that which is formal, formally formal” what does it mean?
Nov
28
comment Relation-preserving maps as morphisms of a category
You seem to be missing a few words after Kleisli category. Perhaps you mean something like "the Kleisli category of the covariant powerset monad"?
Nov
28
comment Question about computability of true/provable formulas
$T_\text{tr}$ cannot possibly be recursively enumerable. If it were we could solve the halting problem: because, if $\phi$ is the assertion that some Turing machine $\mathfrak{M}$ halts, then either $\phi$ is in $T$ or $\lnot \phi$ is in $T$; so we could construct a Turing machine that searches for either $\phi$ or $\lnot \phi$ in $T_\text{tr}$, and this is guaranteed to halt.
Nov
28
comment The ideal of a point
There's absolutely nothing wrong with changing coordinates so that $x = (0, \ldots, 0)$. If you don't like that, you can just go through the proof of the special case and replace $T_i$ with $T_i - x_i$ at every step.
Nov
28
comment Why isn't this free product of groups abelian?
The claim is obvious for free groups: the free product of free groups is again a free group, and any free group on more than one generator is non-abelian. Otherwise some work is needed: the easiest way to proceed is to find a pair of homomorphisms $A_1 \to S$, $A_2 \to S$, $S$ non-abelian, such that the subgroup generated by the union of the images of $A_1$ and $A_2$ is non-abelian.
Nov
28
comment Why isn't this free product of groups abelian?
Your claim as stated is false: $1 * 1 \cong 1$, and $1$ is certainly abelian.
Nov
26
revised Inductively and Co-Inductively defined sets
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Nov
26
revised Inductively and Co-Inductively defined sets
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Nov
26
revised Inductively and Co-Inductively defined sets
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Nov
26
answered Inductively and Co-Inductively defined sets
Nov
26
answered The empty set in homotopy theoretic terms (as a simplicial set/top. space)
Nov
26
answered Kan fibrations and surjectivity