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Mar
17
answered Paths between 0-cells in a classifying space.
Mar
17
comment Examining every mathematical result in purely formal, ZFC language.
Writing rigorous proofs does not mean formulating it in terms of set theory and proving it from the axioms of ZFC. For one thing, the explicit statement in the language of set theory would be nigh unrecognisable to the vast majority of mathematicians...
Mar
16
comment Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem
A quasicoherent sheaf does not correspond to a vector bundle, in general: for one thing, quasicoherent sheaves can be infinite-dimensional, and for another, the dimension of a fibre can change from point to point. The pushforward is not easy to describe in terms of bundles, but in terms of sections, given a continuous map $f : X \to Y$, the pushforward sheaf $f_* \mathscr{F}$ is the sheaf on $Y$ such that the sections of $f_* \mathscr{F}$ over an open set $V$ are the same as the sections of $\mathscr{F}$ over $f^{-1} V$.
Mar
16
comment Alternative set theories
@AndresCaicedo In the last paragraph, did you perhaps mean "Mathias" instead of "Mac Lane"?
Mar
16
comment Alternative set theories
There is an article on Randall Holmes's webpage with the same title.
Mar
16
comment Forgetful Functors Create Limits
It comes from the exercises here.
Mar
16
comment Did large cardinals exist before 1963?
However, were they known at the time, or at least suspected, to be large cardinals in the sense of being cardinals whose existence is not provable?
Mar
15
answered Pure Submodules and Finitely Presented versus Finitely Generated Submodules
Mar
15
comment Pure Submodules and Finitely Presented versus Finitely Generated Submodules
Because every module is a directed limit of finitely-presented modules.
Mar
14
answered Does the inclusion from affine schemes into schemes preserve pushouts?
Mar
14
comment Equivalence of categories of coalgebras
If $R$ is monadic then you may as well assume $X = Y^M$, but that does somewhat trivialise the question. So perhaps what you should do is show that $C$ and $C'$ are equivalent in a suitable sense.
Mar
13
comment Does the term consistency (for equations) have some logical meaning?
One has to be a little bit careful about what one means by ‘logical inconsistency’ in the context of algebraic theories. For example, when a system of polynomials has no solution, the system is inconsistent – in the sense that every other equation (including $0 = 1$) can be derived from it. (Of course, you could say that $0 \ne 1$ is part of the logical axioms and so derive $\bot$, but I prefer to work in the positive fragment.)
Mar
13
comment Forgetful Functors Create Limits
Yes, you could do it like that if you want. Alternatively, note that you have a commutative triangle of forgetful functors, all of which preserve limits, and two of them create limits.
Mar
13
comment Scheme glued out of spectra of local rings
@LokiClock Yes, the category of affine schemes is more-or-less $\textbf{CRing}^\textrm{op}$, but $\textbf{Sch}$ is a strictly larger category (and so limits/colimits can change).
Mar
13
comment Scheme glued out of spectra of local rings
@RonaldBernard The two propositions are not equivalent. $\operatorname{Spec}$ does not always map limits to colimits.
Mar
13
comment Forgetful Functors Create Limits
If it creates limits, then the underlying set of $M$ is $L$. So all you have to do is put the appropriate algebraic structure on $L$. (If widgets are too bizarre for your taste, try doing it for groups first.)
Mar
13
comment Properties of the Category of topological spaces with $n$ basepoints.
Isn't $\textrm{id} : B \to B^\textrm{indisc}$ the terminal object in $\textbf{Top}_B$?
Mar
13
comment What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$?
Merely knowing that $\textbf{Set}_{/ X}$ is a topos is not very useful though. It's more important to note that the étale geometric morphism $\textbf{Set}_{/ X} \to \textbf{Set}$ has a logical inverse image functor.
Mar
13
awarded  category-theory
Mar
12
answered What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$?