added the definition to the old stub *t-structure*. Also redicrecting *heart*, of course. And also for stable $\infty$-categories.

I’ve been working on spectral sequence. But not done yet.

]]>started an index of the keywords in Ravenel’s book *Complex cobordism and stable homotopy groups of spheres*.

So far I got through most of the first chapter. Touched many of the entries involved.

The main entries *chromatic spectral sequence* and *EHP spectral sequence* are stubs for the moment.

consider the Koszul-Tate resolution, which is a dgcalg.

Beyond their both being resolutions, what more can be said? ]]>

I have finally filled content into the entry *derived functor in homological algebra*.

That entry had existed in template form for years, with the intention to eventually take up that content, but clearly I had forgotten to actually put it there after I had written it out on my own web at *HAI (schreiber)*. Now I have copied it over.

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.

Is the set of natural transformations between two simplicial (abelian) groups a simplicial group or a chain complex?

]]>Is there a well defined internal hom for cosimplicial objects? (sets, algebras, rings)

]]>(http://ncatlab.org/nlab/show/Eilenberg-Zilber+map)

(http://ncatlab.org/nlab/show/Alexander-Whitney+map)

are given in the "standard simplicial dimension notation". However in the setting of abelian simplicial groups

and chain complexes we have frequently the situation where we work with augmented simplicial sets

and in that scenario there is the 'upshifted dimension counting' where we define the dimension of the augmented

simplex as zero instead of $(-1)$. (Explained for example in

http://ncatlab.org/nlab/show/simplex+category

My question is now how this affects the definition of the above maps and since I can't find anything on the web

I suggest to add such a augmented definition to the nLab entries on those topics.

If someone can post a link or something where this is worked out, I will change the entry if you people agree. ]]>

Given a simplicial abelian group, the alternating sum defines a derivation, making the simplicial abelian group itself into a chain complex.

The derivation is then an endomorphism on that chain complex. But the complex has another point of view because it is still a simplicial set, too and the question is:

Does the alternating-sum-derivation respect the simplicial structure, i. e. does it commute with the face and degeneracy maps? (Maybe not because of the square to zero rule, but anyway …)

If not, is there a derivation respecting the simplicial structure?

]]>(First, an apology: I accidentally posted this in the wrong place, under “MathForge general discussions.” I tried to delete it from there but couldn’t figure out how to do so. Could someone help me?)

My question relates to the entry

http://ncatlab.org/nlab/show/derived+functor#InHomologicalAlgebra

I’d like to say that $RCh_*(F)$ is a point-set right derived functor of $Ch_*(F)$ but I don’t know how one knows that there is a fibrant replacement *functor* on $Ch_*(A)$ (here meaning the category of non-negatively graded cochain complexes), assuming $A$ has enough injectives.

Naively of course, one can construct an injective resolution $I_*(A)$ of an object of $A$. Given a map $A \to B$ one can build a map $I_*(A) \to I_*(B)$ but this is only unique up to chain homotopy equivalence and so this construction is not functorial on the point-set level (though of course suffices for defining total derived functors). Presumably this naive construction can be generalized to construct, given a non-negative graded cochain complex $A_*$, a cochain complex $I_*(A_*)$ of injectives together with a quasi-isomorphism $A_* \to I_*(A_*)$ but will similarly fail to be strictly functorial.

In the dual situation (non-negatively graded chain complexes) if we take the abelian category $A$ to be modules over some ring, then the appropriate model structure is cofibrantly generated so it’s clear there is a functorial cofibrant replacement. But I don’t know that the injective model structure is cofibrantly generated, even in this case.

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