I have reorganised set theory and spun off material set theory.

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

]]>The stub entry *model structure on simplicial Lie algebras* used to point to *model structure on simplicial algebras*. But is it really a special case of the discussion there?

Quillen 69 leaves the definition of the model structure to the reader. Is it with weak equivalences and fibrations those on the underlying simplicial sets? Is this a simplicially enriched model category?

]]>started a Properties-section at Lawvere theory with some basic propositions.

Would be thankful if some experts looked over this.

Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.

]]>for the purposes of having direct links to it, I gave a side-remark at *stable Dold-Kan correspondence* its own page: rational stable homotopy theory, recording the equivalence

I also added the claim that under this identification and that of classical rational homotopy theory then the derived version of the free-forgetful adjunction

$(dgcAlg^{\geq 2}_{\mathbb{Q}})_{/\mathbb{Q}[0]} \underoverset {\underset{U \circ ker(\epsilon_{(-)})}{\longrightarrow}} {\overset{Sym \circ cn}{\longleftarrow}} {\bot} Ch^{\bullet}(\mathbb{Q})$models the stabilization adjunction $(\Sigma^\infty \dashv \Omega^\infty)$. But I haven’t type the proof into the entry yet.

]]>split off model structure on simplicial T-algebras from (infinity,1)-algebraic theory

]]>Has anyone developed models for the homotopy theory of $H \mathbb{Q}$.module spectra over rational topological spaces a bit?

I expect there should be a model on the opposite category of dg-modules over rational dg-algebras. Restricted to the trivial modules it should reduce to the standard Sullivan/Quillen model of rational homotopy theory. Restricted to the dg-modules over $\mathbb{Q}$ it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of $H \mathbb{Q}$-module spectra.

Is there any work on this?

]]>I gave *simplicial Lawvere theory* an entry, stating Reedy’s result on the existence of the simplicial model structure of simplicial algebras over a simplicial Lawvere theory

almost missed that meanwhile we have an entry *pullback-power*. So I added more redirects and expanded a little.

Added the example of smooth manifolds, which have a canonical fully faithful embedding into locally ringed spaces, citing Lucas Braune’s nice proof on stackexchange.

]]>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 gave *continuous map* a little bit of substance by giving it an actual Idea-paragraph and by writing out the epsilontic definition for the case of metric spaces, together with its equivalence to the “abstract” definition in terms of opens.

Fixed the comments in the reference list at [[model structure on dg-algebras]]: Gelfand-Manin just discuss the commutative case. The noncommutative case seems to be due to the Jardine reference. Or does anyone know an earlier one?

]]>polished regular category somewhat and added a few things, for instance a Properties-section.

Also started and linked to a dedicated entry on regular logic.

]]>The entry *minimal fibration* used to be just a link-list for disambiguating the various versions. I have now given it some text in an Idea-section and a pointer to Roig 93 where the concept is considered in generality.

I expanded [[proper model category]] a bit.

In particular I added statement and (simple) proof that in a left proper model category pushouts along cofibrations out of cofibrants are homotopy pushouts. This is at Proper model category -- properties

On page 9 here Clark Barwick supposedly proves the stronger statement that pushouts along all cofibrations in a left proper model category are homotopy pushouts, but for the time being I am failing to follow his proof.

(??)

]]>https://ncatlab.org/jamesdolan/published/Algebraic+Geometry ]]>

Somebody kindly pointed out by email to me two mistakes on the page Pr(infinity,1)Cat. I have fixed these now (I think).

The serious one was in the section Embedding into Cat where it said that $Pr(\infty,1)Cat \to (\infty,1)Cat$ preserves limits and colimits. But it only preserves limits. This is HTT, prop. 5.5.3.13. The wrong statement was induced from a stupid misreading of HTT, theorem. 5.5.3.18. Sorry.

The other mistake was that it said “full subcategory”. But of course by the very definition of $Pr(\infty,1)Cat$ if is not full in $(\infty,1)Cat$. I have fixed that, too, now.

]]>I was dissatisfied with the discussion at semisimple category because it only defined a semisimple *monoidal Vect-enriched* category, completely ignoring the more common notion of semsimple abelian category.

So, I stuck in the definition of semisimple abelian category.

However, I still think there is a lot that could be improved here: when is a semisimple abelian category which is also monoidal a semsimple monoidal category in some sense like that espoused here???

I think this article is currently a bit under the sway of Bruce Bartlett’s desire to avoid abelian categories. This could be good in some contexts, but not necessarily in all!

]]>Created sound doctrine as a stub to record relevant references.

]]>I came to think that the pattern of interrelations of notions in the context of locally presentable categories deserves to be drawn out explicitly. So I started:

Currently it contains the following table, to be further fine-tuned. Comments are welcome.

| | | inclusion of left exaxt localizations | generated under colimits from small objects | | localization of free cocompletion | | generated under filtered colimits from small objects |
|–|–|–|–|–|—-|–|–|
| **(0,1)-category theory** | (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | | |
| **category theory** | toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories |
$\hookrightarrow$ | accessible categories |
| **model category theory** | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | | |
| **(∞,1)-topos theory** | (∞,1)-toposes |$\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ <br/> Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories |
$\hookrightarrow$ |accessible (∞,1)-categories |

For a while, we've been blanking spam pages and putting them in their own category; then people are supposed to use these when they create a new page. This effectively removes the spam page from circulation without having to invoke deletion. Then we found some other junk pages that weren't spam but either mistakes or whatever is going on with those weird slideshow pages. Then we put some of these in category: empty since they weren't spam. But we remarked that they really could all go there.

So now I've put them all there and regularised the names.

Please use these when you make a new page! Only a few people have to remember to do this and they will all go away. (Zoran remembers, but I usually forget.)

]]>