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Apr
12
comment Direct cut-elimination for natural deduction.
Do you mean, "the analogue of cut elimination in natural deduction"?
Apr
11
comment A reference that forcing with a poset of cardinality less than $\kappa$ preserves supercompact cardinals
This seems to be on page 390 in the Third Millennium edition.
Apr
11
comment coequalizers+pullbacks implies equalizers?
Clever! This amounts to the observation that the projection $\mathcal{C}_{/ C} \to \mathcal{C}$ preserves equalisers and creates pullbacks, and $\mathcal{C}_{/ C}$ has products if $\mathcal{C}$ has pullbacks.
Apr
11
revised Elephant: how do I prove Lemma 2.1.7, section C2.1?
smaller counterexample
Apr
11
answered Elephant: how do I prove Lemma 2.1.7, section C2.1?
Apr
11
comment Does there exist a topology for a set $X$ which is compact and Hausdorff?
Crossposted at MO.
Apr
11
comment Elephant: how do I prove Lemma 2.1.7, section C2.1?
It would be good if you also stated the lemma, or at the very least, what $s_{i,j}$ and $s_i$ are.
Apr
11
revised Elephant: how do I prove Lemma 2.1.7, section C2.1?
edited tags
Apr
11
comment Is the class of all ordinals independent of set theory?
Why does Löwenheim–Skolem imply there are countable transitive models? Is it not possible that there are only uncountable transitive models?
Apr
11
comment Functions and metafunctions
No internal function is a metafunction and no metafunction is an internal function, because the domain of an internal function is a set inside the universe, whereas the domain of a metafunction is the entire universe. But this is only a technicality; otherwise every internal function gives rise to a unique partially-defined metafunction, and a metafunction restricted to a set gives rise to an internal function (by replacement).
Apr
10
comment Calculating de Rham Comohologies of Punctured Manifolds
Actually, number of path components would be more accurate if we are working in the generality of all topological spaces...
Apr
9
comment Accessible introduction to category theory from the point of view of preorders.
How about Locally presentable and accessible categories? It even has the word ‘accessible’ in the title! (I jest!)
Apr
9
comment How does $C$ small imply $Set^{C^{op}}$ locally small?
@almaus I know plenty of categories that are not locally small. Some people choose not to accept this.
Apr
9
comment How does $C$ small imply $Set^{C^{op}}$ locally small?
@Berci There are people who don't believe in categories that are not locally small...
Apr
8
comment How does the Yoneda lemma imply that $\mathrm{Hom}(yC,P)$ is a set?
It doesn't help, unless you are dead-set on making it literally true that $\textrm{Hom}(y C, P)$ is a set. (Every set is in bijection with an von Neumann ordinal but not in a canonical way, so it is difficult to identify sets with ordinals.)
Apr
8
comment How does the Yoneda lemma imply that $\mathrm{Hom}(yC,P)$ is a set?
For the reasons Martin has highlighted, it may be better to say "is small" or "is in bijection with a set" instead of "is a set".
Apr
8
comment How does the Yoneda lemma imply that $\mathrm{Hom}(yC,P)$ is a set?
@Zev It depends on the precise details of how $\textrm{Hom}(y C, P)$ is coded, but Martin's claim is correct in any sensible coding where natural transformations are coded as something that contains their graph. The bottom line is, a set is something that is hereditarily small. For example, if $U$ is the universe of all sets, then the collection $\{ U \}$ is not a set because one of its members is not a set.
Apr
8
comment Does the statement “There is an algorithm that solves …” make sense?
I think the crux of the issue is uniformity. Of course there is an algorithm for each $(a, b)$ that solves the problem $P(a, b)$ in $O(1)$; the real problem is, does there exist one algorithm that solves $P(a, b)$ for all $(a, b)$?
Apr
8
comment How exactly does a map represent an operation?
It should be "contravariant", not "covariant".
Apr
8
comment How exactly does a map represent an operation?
You can read "represents" naïvely here in the English sense, but it also "represents" in the technical sense of representable functors. (Recall that the object $2$ is a subobject classifier for $\textbf{Set}$.)