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Jul
26
comment topos have colimits
You should learn about monads. The concrete details are painful even in the case of the initial object: see here.
Jul
26
comment coherence of inverses in 2-groupoids
1-cell inversion acts on 2-cells. In fact, the natural operation sends a 2-cell $f \Rightarrow g$ to a 2-cell $g^{-1} \Rightarrow f^{-1}$.
Jul
25
comment Restriction of a sheaf equals zero
Most likely, these are sheaves of abelian groups, in which case a sheaf is zero if every section is zero.
Jul
25
comment Is direct limit of local rings a local ring?
Yes, it is a local ring.
Jul
24
comment coherence of inverses in 2-groupoids
It seems to me that your operation is the composite of 2-cell inversion and 1-cell inversion.
Jul
23
comment Fiber bundles with category morphisms as fibers
Your question is missing many details. In what sense are the elements of your fibres morphisms? What are your "transfer functions" supposed to achieve? Why are you suddenly able to differentiate when you were not able to before? etc.
Jul
23
comment Intuition on the Representable Functor
It is just a name. We also say that $C$ represents the functor $\mathrm{Hom} (C, -)$.
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
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
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 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.