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Dec
19
accepted When is $N\otimes_A B \to N$ an isomorphism?
Dec
15
awarded  Self-Learner
Nov
30
comment Will learning 'classical geometry' a la Hilbert's incedence geometry help me appreciate the results of classical algebraic geometry?
I'd like to iterate the question: will learning classical algebraic geometry help with modern algebraic geometry (following Grothendieck's school) or even derived algebraic geometry? I feel like without the enlightening notion of sheaf and without the powerful tools of commutative algebra, one could easily get lost in the deceptive intuition provided by classical algebraic geometry.
Nov
16
comment Are split exact sequences exact in the opposite direction?
@Exterior If $f, g, r, \ell$ form a biproduct diagram (what Freyd calls direct sum system, see §2.4), then $(r, \ell)$ is exact basically by definition. Freyd's Proposition 2.42 is actually a necessary condition, as well; thus in order for our four morphisms to be a biproduct diagram, we need both the sequences of the OP to be exact. Furthermore, Proposition 2.68 tells us that for a given $\ell$ there exists $r$ such that these give rise to a biproduct diagram; moreover, as $r$ must be the kernel of $\ell$, this is uniquely determined up to automorphism of $C$.
Nov
14
awarded  Revival
Nov
10
revised When is $N\otimes_A B \to N$ an isomorphism?
forgot a $
Nov
10
answered When is $N\otimes_A B \to N$ an isomorphism?
Oct
15
comment Are all fibres of a trivial fibre space over X with fibre F canonically homeomorphic with F?
Without assuming some coherence among the topology $\tau$ on $F$ and the one, say $\sigma$, on the product $X\times F$, it's really easy to build counter-examples. For instance, suppose $F$ has at least two elements and take $X = \{\emptyset\}$ (the singleton with its unique topology) and put on $X\times F \cong F$ a topology $\sigma$ different from $\tau$ (the initial and terminal topology, for example). The fiber is then homeomorphic to $(F, \sigma)$ and thus not homeomorphic to $(F, \tau)$
Oct
14
comment Why is the Quillen model structure so painful to find?
I agree with @ZhenLin on both points. I'd guess May is referring to some classical transfer result (as described here). Further, the class of weak equivalences of the Kan-Quillen model structure on simplicial sets can be described intrinsically as the $\Delta$-localizer generated by the projections $\Delta_1 \times X \rightarrow X$, for any presheaf $X$ on $\Delta$.
Oct
10
awarded  Nice Question
Apr
8
comment Is there a characterization of coverings in subcanonical pretopologies?
Nice! Do you have a reference?
Apr
2
comment Filtered colimits are exact in abelian categories
The exactness of filtered colimits is exactly the axiom (AB5) (given by A. Grothendieck in his famous Tōhoku paper). There are a couple of interesting remarks about this axiom: an equivalent form of (AB5), which is worked out magnificently here, and the incompatibility with axiom (AB5*), which gives you many examples of abelian categories which do not satisfy (AB5), $\mathbf{Ab}^{\text op}$ in particular.
Mar
16
comment A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
The axiom (AB4) ensures the commutativity of homology with coproducts. A more general result, from which this exercise follows as corollary, is proven in Lemma 12, Section III.7.13 of Methods of Homological Algebra by S. Gelfand and Y. Manin.
Mar
10
comment A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
Why don't you try to use a spectral sequence for this double complex? The canonical filtration for rows should degenerate at sheet 2, giving you exactly the isomorphisms between the homologies you're looking for.
Mar
10
comment Are split exact sequences exact in the opposite direction?
If you have the first short exact sequence and $\ell f = 1_A$, then you can choose $r$ in such a way $gr= 1_C$ and the second sequence is short exact. This is Proposition 2.68 "Splitting maps" of Abelian Categories by P. Freyd.
Feb
25
comment Associativity of the tensor product of dendroidal sets
A complete and nice answer by G. Heuts can be found here.
Feb
25
revised Associativity of the tensor product of dendroidal sets
Miscalculations corrected
Feb
20
comment Associativity of the tensor product of dendroidal sets
I copied the same thread on MO, as well.
Feb
20
revised Associativity of the tensor product of dendroidal sets
Added more background
Feb
20
revised Associativity of the tensor product of dendroidal sets
typo