added some basics to *model structure for quasi-categories* at *general properties*

I have given *Grothendieck construction for model categories* its own entry, in order to have a place for recording references. In particular I added pointer to the original references (Roig 94, Stanculescu 12)

(There used to be two places in the entry *Grothendieck construction* where an attempt was made to list the literature on the model category version, but they didn’t coincide and were both inclomplete. So I have replaced them with pointers to the new entry.)

I am splitting off *homotopy category of a model category* from *model category*. Have spelled out statement and proof of the localization construction there.

created *Homotopy Type System* with an Idea-section

have started model structure for L-infinity algebras

]]>I needed an entry for Charles Rezk’s term *model topos* in order to complete *locally presentable categories - table*, and so I created it. But we should have had that anyway. I have also cross-linked it with relevant entries

created *fibrant type* with an Idea-section

created Cisinski model structure

]]>I have split off *classical model structure on topological spaces* from the entry on “model structures on topological spaces”.

My aim is to have in this entry a detailed, self-contained and polished account of the definition of the *standard* or *classical* model structure, its verification and its key consequences.

I have added a fair bit of material today. Not done yet, but I have to call it quits now.

]]>I added two characterisations of weak homotopy equivalences to model structure on simplicial sets.

For the record, I found the inductive characterisation in Cisinski’s book [*Les préfaisceaux comme modèles des types d’homotopie*, Corollaire 2.1.20], but I feel like I’ve seen something like it elsewhere. The characterisation in terms of internal homs comes from Joyal and Tierney [*Notes on simplicial homotopy theory*], but they take it as a *definition*.

Is the Strøm model category left proper? I know that pushout along cofibrations of homotopy equivalences of the form $A \to \ast$ are again homotopy equivalences. (e.g. Hatcher 0.17) Maybe the proof directly generalizes, haven’t checked.

]]>I added an explicit definition of cartesian model category to cartesian closed model category to highlight the convention that the terminal object is assumed cofibrant.

]]>I have split off an entry *homotopy in a model category* from *homotopy* and then spelled out statement and proof of the basic lemmas.

I have split off an entry *classical model structure on simplicial sets* from “model structure on simplicial set”. This entry should eventually contain detailed, self-contained and polished discussion of the definition, verification and key properties of the standard Kan-Quillen model structure.

So far I have inserted fair bit of background material regarding (minimal) fibrations and geometric realizations, essentially the material in chapter 1 of Goerss-Jardine. A bunch of little proofs are spelled out, but not yet the more laborious ones. Discussion of the verification of the axioms is not yet in the entry, but the key parts of the Quillen equivalence to $Top_{Quillen}$ are (modulo relying on previous lemmas that don’t have proofs spelled out yet).

The somewhat random list of properties of $sSet_{Quillen}$ that used to be sitting at “model structure on simplicial sets” I have copied over to a section “Basic properties”, just for completenes, but this now needs re-organization to give decent logical flow.

For the moment I have to leave it at that, need to take care of something else now for a little bit.

]]>At *projective resolution* I have

spelled out the

*Definition*in lots of detail;spelled out statement and proof of the

*existence of resolutions*in full detail.

I have added some basic content to the old entry *fibrant object*: An idea-section, the statement about (co)fibrant replacement, and a few basic examples.

created *model structure on dg-comodules*, just so as to record a pointer to Positelski 11, theorem 8.2.

Regarding the dg-comodules which are injective as graded comodules over the underlying graded cocommutative co-algebra: Suppose the latter is co-free and the ground ring is a field. Is it then true that all injective comodules over it are cofree? Because this would seem to be a dual version of the Quillen-Suslin theorem?

]]>(never mind)

]]>stub for model structure on dg-Lie algebras

]]>am starting [[model structure on dg-coalgebras]].

In the process I

created a stub for [[dg-coalgebra]]

and linked to it from [[L-infinity algebra]]

I have created an entry *model structure on topological sequential spectra*.

In parts this directly parallels the entry *Bousfield-Friedlander model structure*.

But now I have spelled out full proof of the model structure and its cofibrant generation: here

I did this by taking the more general proof that I had earlier spelled out at *Model categories of diagram spectra*, and specializing it to the case of sequential spectra.

The effect of that is that those tedious technical lemmas about the maps of free spectra collapse to something simple, with the result that the actual proof may start right away with less preliminaries, which makes the writeup a bit more transparent. On the other hand, the neat thing is that apart from that analysis of the free spectra the proof is verbatim the same now for all cases (sequential, symmetric, orthogonal spectra and pre-excisive functors), so in the other entries it’ll be possible to turn this around and say: “after this analysis of the free symmetric/orthogonal spectra the proof of their model structure now follows verbatim as at model structure for topological sequential spectra”.

As far as exposition and writeup goes, the only remaining “gap” I left is that at one point the proof invokes that $Top_{Quillen}^{\ast/}$ and hence $[StdSpheres, Top_{Quillen}^{\ast/}]_{proj}$ is a topological model structure (this is used in the proof of this lemma ). I plan to spell that out, too. But not tonight.

]]>started something at *Bousfield-Friedlander model structure*

I have added to *symmetric spectrum* (after the definition in components) also discussion of the definition as $\mathbb{S}_{Sym}$-module objects with respect to Day convolution over $Core(FinSet)$ (here).

I am really in the middle of some editing here, but need to call it quits for tonight.

]]>collected some references at *model structure for n-excisive functors* and added cross links.

I had not been aware before that Lydakis also got a symmetric monoidal smash on the model structure for excisive functors.

]]>I found that the $n$Lab was and is a bit weak on topics related to highly structured ring spectra. We did (and do) have a nice entry *symmetric monoidal smash product of spectra*, but I think for a decent coverage we need more.

I started out adding stuff to the following circle of entries.

model structure on spectra, symmetric monoidal smash product of spectra

But the nLab doesn’t like being edited so much and gives me a hard (down-)time. Makes me run out of steam a bit. So much of this remains super-stubby for the moment. But we should eventually expand…

]]>