nForum - Search Results Feed (Tag: homology) 2021-09-24T02:09:48-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher A correction. https://nforum.ncatlab.org/discussion/8291/ 2018-02-18T04:38:11-05:00 2018-02-18T04:38:11-05:00 Murat_Aygen https://nforum.ncatlab.org/account/1558/ Theorem: Let KK be a complex with rr combinatorial components. Then H 0(K)H_{0}(K) is isomorphic to the direct sum of (not rr but) 2r2r copies of the group ZZ of integers. Proof for one ...

Theorem:

Let $K$ be a complex with $r$ combinatorial components. Then $H_{0}(K)$ is isomorphic to the direct sum of (not $r$ but) $2r$ copies of the group $Z$ of integers.

Proof for one component:

As 0-cycles are synonymous with 0-chains, they have $\alpha_{0}$ many dimensions-of-freedom for sure. How many the 0-boundaries of our component have? Since the incidence matrix is a $\alpha_{0}$ by $\alpha_{1}$ matrix (see my previous posting with ‘homology’ tag), they have at least $\alpha_{1} - \alpha_{0}$ many for sure. And 1 more for the rank deficiency! The proof follows from this arithmetic.

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Incidence matrix https://nforum.ncatlab.org/discussion/8287/ 2018-02-14T08:45:26-05:00 2018-02-14T08:45:26-05:00 Murat_Aygen https://nforum.ncatlab.org/account/1558/ If one solves b1⟨a1⟩+…+bm⟨am⟩=δ[∑∑xi,j⟨ai,aj⟩] for x’s when b’s are given, one ends up with ηx→=b→ where \eta is the INCIDENCE MATRIX! Then all proofs about 0-homology ...

If one solves

${b}_{1}⟨{a}_{1}⟩+\dots +{b}_{m}⟨{a}_{m}⟩=\delta \left[\sum \sum {x}_{i,j}⟨{a}_{i},{a}_{j}⟩\right]$

for x’s when b’s are given, one ends up with

$\mathbit{\eta }\stackrel{\to }{x}=\stackrel{\to }{b}$

where \eta is the INCIDENCE MATRIX! Then all proofs about 0-homology groups are just about the ranks of square-submatrices of this matrix. Am I right?

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Kernel of split epimorphism is cokernel of right-inverse? https://nforum.ncatlab.org/discussion/4965/ 2013-06-01T01:55:13-04:00 2013-06-13T06:25:32-04:00 joe.hannon https://nforum.ncatlab.org/account/887/ In reduced homology#relation to relative homology, at the bottom is a computation where one step is that kerH 0(&epsi;)&cong;cokerH 0(x).\ker H_0(\epsilon) \cong \operatorname{coker}{H_0(x)}. ...

In reduced homology#relation to relative homology, at the bottom is a computation where one step is that $\ker H_0(\epsilon) \cong \operatorname{coker}{H_0(x)}.$ Can you explain that step to me? If you think of kernels and cokernels as morphisms, then there’s no way such an isomorphism can hold, since $\ker H_0(\epsilon)$ is a morphism into $H_0(X)$, while $\operatorname{coker}{H_0(x)}$ is a morphism out of $H_0(X).$ But if you think of kernels and cokernels as objects, then I guess it’s possible for them to be isomorphic. My guess is it must follow from the fact that $H_0(\epsilon)\circ H_0(x)$ is an iso, so the statement is something like “the kernel of a split epimorphism is the cokernel of its right-inverse,” but I can’t figure it out.

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Homology from the nPOV https://nforum.ncatlab.org/discussion/2214/ 2010-12-01T05:59:19-05:00 2010-12-03T11:09:30-05:00 Mike Shulman https://nforum.ncatlab.org/account/3/ If generalized nonabelian cohomology, from the nPOV, means a hom-space in some (&infin;,1)(\infty,1)-topos, then it can equivalently be characterized as global sections of an object in some ...

If generalized nonabelian cohomology, from the nPOV, means a hom-space in some $(\infty,1)$-topos, then it can equivalently be characterized as global sections of an object in some $(\infty,1)$-topos, since for any $X,A$ in an $(\infty,1)$-topos $E$ we have $Hom_E(X,A) = Hom_{E/X}(1,X^*A) = \Gamma_{E/X}(X^*A)$. Recall that traditional “abelian” sheaf cohomology $H^n(X;A)$ is the case when $A$ is an $n$-fold delooping of a discrete abelian group object, and when $A$ is locally constant (whatever that means) it reduces to “cohomology with local coefficients” and further to the most traditional algebraic-topology sort of cohomology when $A$ is constant. Generalizing in a different direction, if $A$ is constant, not on a discrete abelian group, but on a spectrum, then we obtain classical “generalized cohomology”, and if we further generalize to spectrum objects in $E$ then we have “generalized sheaf cohomology” with coefficients in a sheaf of spectra. Note that $\Gamma$ preserves abelian group objects and spectrum objects, so that with these definitions, abelian cohomology theories always produce abelian objects.

In another thread I asked a question about homology from the nPOV (and David C kindly supplied some links to past discussions). A couple of answers were given, but I just thought of a slightly different way of stating it, which I like. Suppose our $(\infty,1)$-topos is locally ∞-connected, so that in addition to a right adjoint $\Gamma$, the constant stack functor $\Delta$ has a left adjoint $\Pi$. Now $\Pi$ won’t preserve abelian and spectrum objects, but by general “adjoint lifting theorems” I would nevertheless expect to be able to construct from it a functor $\Pi_{spectra}: Spectra(E) \to Spectra$ which is left adjoint to $\Delta_{spectra}: Spectra \to Spectra(E)$. It seems to me that it would make sense to regard $\Pi_{spectra}$ as a notion of “sheaf homology” with coefficients in a sheaf of spectra (perhaps a constant one).

It could be that this is way off-base, but I’m getting my intuition from the May-Sigurdsson theory of “parametrized” spectra, which should morally (I believe) be identifiable with “locally constant” sheaves of spectra over nice spaces. In their context, the pullback functor $r^* : Spectra \to Spectra/X$ always has both a left adjoint $r_!$ and a right adjoint $r_*$, and the left adjoint is homology while the right adjoint is cohomology. In particular, $r_! r^* M$, for a spectrum $M$, can be identified with the generalized homology theory $H_*(X;M)$.

Thoughts? Is this obviously true? Obviously false?

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