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Apr
28
comment On the large cardinals foundations of categories
Yes, and even better, we have a filtration of the whole universe by these weak universes, since every $V_\lambda$ for limit ordinals $\lambda > \omega$ is a weak universe. But I am not so willing to assert that weak universes are good enough for category theory – it would require too much checking to make sure that someone didn't secretly use replacement somewhere...
Apr
28
comment On the large cardinals foundations of categories
No, what I call a ‘weak universe’ is basically a "intended model" of Zermelo set theory. In fact, unless I'm mistaken, it is precisely the same thing as a transitive model of Zermelo set theory that has the same powersets and $\omega$ as the ambient set theory.
Apr
28
comment On the large cardinals foundations of categories
Well, people who believe ETCS is adequate for mathematics should also believe that weak universes (as in ¶ 4.2) are adequate for developing category theory. The consistency strength of a weak universe axiom is at most that of ZFC.
Apr
28
comment On the large cardinals foundations of categories
See the introduction and the last paragraph of the last section of my article here. The short answer is that moving between mere models of ZFC is very hard using the technology of category theory, but moving between Grothendieck universes is fairly tractable.
Apr
28
revised Defining Test-Objects
added 29 characters in body
Apr
28
comment Category theory $\subset$ Set theory or vice versa?
You can do quite a bit of pure category theory within the confines of ordinary set theory (by focusing on category-theoretic analogues of what is studied in a first course about rings and modules, say), but for most applications one wants to be able to talk about categories whose collection of objects is "large" in some sense.
Apr
28
answered Defining Test-Objects
Apr
27
comment Defining $\Bbb{Q}$ without the axiom of infinity
You could always represent rational numbers the same way as schoolchildren: as reduced fractions.
Apr
27
comment Sub-(complete lattice)? - what's the correct terminology?
That's right: frame homomorphisms are not the same as homomorphisms of complete Heyting algebras.
Apr
26
comment pullback square of regular epimorphisms is a pushout
See Proposition 1.4.3 in [Johnstone, Sketches of an elephant, Part A].
Apr
26
comment Connection between the limiting cones
(Co)finality is about colimits, however.
Apr
26
revised Connection between the limiting cones
edited body
Apr
26
comment Connection between the limiting cones
If $T$ is coinitial (or initial, according to some authors) then it has properties 1 and 2.
Apr
26
comment Sub-(complete lattice)? - what's the correct terminology?
Along similar lines, it is very bad form to conflate frames and complete Heyting algebras. The data of a complete Heyting algebra includes the Heyting operation, which automatically exists in any frame, but a frame homomorphism need not preserve the Heyting operation.
Apr
26
comment Sub-(complete lattice)? - what's the correct terminology?
No: the data of a frame only includes finite meets and possibly-infinite joins, so to be a subframe, you only need to have the same finite meets and possibly-infinite joins. Saying that a poset is "closed" under [operation] is bad terminology: it suggests there is an absolute notion of [operation] in some universal poset, and this is not the case. So a subposet of a frame that is closed under joins need not be closed under meets: it will have meets, but they won't be the same in general.
Apr
26
revised Sub-(complete lattice)? - what's the correct terminology?
added 50 characters in body
Apr
26
answered Sub-(complete lattice)? - what's the correct terminology?
Apr
26
revised Injective Resolutions in $\mathfrak{Ab}(X)$
deleted 10 characters in body; edited tags
Apr
26
comment Injective Resolutions in $\mathfrak{Ab}(X)$
Injective resolutions are not easy things to find; that is why we have Čech cohomology!
Apr
26
revised Theory of promonads
added 113 characters in body