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Jul
23
comment Minimum number of sets required for a good open cover
Here is a crude lower bound for the minimum: by considering the homology of the simplicial complex associated with a good open cover, we see that a non-zero homology group in dimension $n$ implies that the good open cover has at least $n + 1$ elements; but the homology of the simplicial complex associated with a good open cover is isomorphic to the homology of the space itself (when the space is nice enough), so this gives a lower bound for all good open covers.
Jul
22
comment Left Kan extension of a $\mathsf{Set}$-valued finite-product-preserving functor
I don't think you will be able to show that the category of elements in question is filtered. At any rate, it would suffice to show that it is sifted.
Jul
22
awarded  Notable Question
Jul
20
comment Definition of Category
Actually, comprehension is somewhat restricted in NBG, at least compared to MK.
Jul
20
comment Unorthodox definition of semi-abelian category
I don't think this is equivalent to the standard definition.
Jul
20
revised On two definitions of the nerve of a simplicial category
edited tags
Jul
20
comment On two definitions of the nerve of a simplicial category
First things first: simplicial sets considered as objects in the Joyal model category must not be confused with simplicial sets considered as objects in the Kan–Quillen model category. The "geometric realisation" of a bisimplicial set is all about the latter, whereas the homotopy coherent nerve is all about the former. Secondly, ordinary categories are too much of a special case to tell you anything interesting here; instead you should be looking at, say, a simplicial monoid considered as a one-object simplicial category.
Jul
20
comment Decidability of equality of two set-theoretical terms constructed without replacement or specification
I think you get a certain amount of separation for free like that – maybe $\Delta_0$.
Jul
18
comment On two definitions of the nerve of a simplicial category
What makes you think the two constructions are related? Just because they have the same name doesn't mean they have to be the same...
Jul
18
comment Are Prevarieties irreducible?
Isn't the union of two intersecting lines connected but not irreducible?
Jul
18
comment Are subvarieties just full subcategories that happen to be algebraic categories?
It is clear that a Birkhoff subcategory of a Birkhoff subcategory is itself a Birkhoff subcategory, so you may certainly assume that $T$ is free.
Jul
18
comment Are subvarieties just full subcategories that happen to be algebraic categories?
Have you looked at Birkhoff's HSP theorem?
Jul
18
comment Continuity of Galois representations from cohomology
I think you would also need to know something about finiteness of cohomology groups.
Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
That's not a boolean algebra, however.
Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
I don't believe that. What's the canonical algebra structure on the $\mathbb{F}_2$-vector space of countable dimension?
Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
It isn't true. Boolean algebras are $\mathbb{F}_2$-algebras but not every $\mathbb{F}_2$-algebra is a boolean algebra. (There's an extra equation.)
Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
No, it is not. Boolean algebras are $\mathbb{F}_2$-algebras.
Jul
16
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
Isn't every finitely generated boolean algebra free?
Jul
16
comment Definition of (left) resolution
I think you are missing intuition. Did you try thinking concretely in terms of abelian groups?
Jul
16
comment Definition of (left) resolution
The proof works both ways, no?