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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• this is a bare little section, to be !include-ed as a Properties-subsection at slice category and adjoint (∞,1)-functor (where two copies of this same section used to be all along) and also at slice (∞,1)-category and adjoint functor (where, for completeness, the same should be recorded, too, but wasn’t until now)

It would also be good to expand a little here, for instance by adding a pointer to a 1-category textbook account (this is probably in Borceux, but I haven’t checked yet), or, of course, by adding some indication of the proof.

• am recording an actual proof that

$\mathcal{L} \big( \overline{W}\mathcal{G} \big) \;\; \simeq \;\; \mathcal{G}_{ad} \sslash \mathcal{G}$

I expected that a proof for this folklore theorem would be citeable from the literature, but maybe not quite. This MO reply points to Lemma 9.1 in arXiv:0811.0771, which has the idea (in topological spaces), but doesn’t explicitly verify all ingredients. I have tried to make it fully explicit (in simplicial sets).

• am starting some minimum here. Have been trying to read up on this topic. This will likely become huge towards beginning of next year

• I added a reference to a paper of mine

Amnon Yekutieli

• Page created, but author did not leave any comments.

• I thought we already long had this as an entry – but, no, it was re-directing all along to model structure on simplicial algebras. Am giving it its own entry now, but it remains telegraphic for the moment.

• added earlier publication items:

• Sören Illman, Equivariant singular homology and cohomology for actions of compact lie groups (doi:10.1007/BFb0070055) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), Proceedings of the Second Conference on Compact Transformation Groups Lecture Notes in Mathematics, vol 298. Springer 1972 (doi:10.1007/BFb0070029)

• Sören Illman, Equivariant algebraic topology, Annales de l’Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 (doi:10.5802/aif.458)

and also more cross-links under “Related entries”

• Someone anonymous has raised the question of subdivision at cellular approximation theorem. I do not have a source here in which I can check this. Can anyone else check up?

• fixed the pointers to alleged proofs of the equivariant Whitehead theorem.

What is a canonical citation for this statement, for the unstable case, citable as an actual proof? Is it due to Bredon? Is it in his book?

• In this entry I mean to write out a full proof for the transgression formula for (discrete) group cocycles, using just basic homotopy theory and the Eilenberg-Zilber theorem.

Currently there is an Idea-section and the raw ingredients of the proof. Still need to write connecting text. But have to interrupt for the moment.

• added to simplicial set in the Definition section a slightly more explicit version of the definition.

(I see now this kind of thing is repeated further below in the entry. But it should be right there as a formal definition, I think.)

• I added to cylinder object a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ a relative cell complex).

What would be a text that features a graphics which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )

• This page did nothing but point to deformation retract. Am have cleared this here now and instead installed a proper redirect

• Since we had bits and pieces of this scattered around over several entries, I am giving this its own little entry, so as to have the statement and all relevant references collected in one place, for ease of updating/synchronizing and hyperlinking.

• giving this its own little entry, for ease of hyperlinking

• Created:

## Idea

Partial model categories are one of the many intermediate notions between relative categories and model categories.

They axiomatize those properties of model categories that only involve weak equivalences.

## Definition

A partial model category is a relative category such that its class of weak equivalences satisfies the 2-out-of-6 property (if $s r$ and $t s$ are weak equivalences, then so are $r$, $s$, $t$, $t s r$) and admits a 3-arrow calculus, i.e., there are subcategories $U$ and $V$ (which can be thought of as analogues of acyclic cofibrations and acyclic fibrations) such that $U$ is closed under cobase changes (which are required to exist), $V$ is closed under base changes, and any morphism can be functorially factored as the composition $v u$ for some $u\in U$ and $v\in V$.

## Properties

If $(C,W)$ is a partial model category, then any Reedy fibrant replacement of the Rezk nerve $N(C,W)$ is a complete Segal space.

## References

• Page created.

• The term simplicial abelian group used to redirect to simplicial group. I am giving it its own little entry now, for better hyperlinking.

• division rigs, the rig version of division rings

Anonymous

• Just a stub for the moment to try to introduce the notion of differential category due to Blute, Cockett and Seely.

• Started writing down the definition.

• Page created, but author did not leave any comments.

• Added some remark on the order of a semiring. Actually, does anybody know if any semiring embedds into a semifield?

• Created this entry with mention of lattice structure and a fundamental theorem of algebra for semifields.

• To fill in a dead link

• have brushed-up the Definition-section

• added redirect plus some ‘selected publications’.

• added redirects for abbreviations of his name, plus some publications. Also added a (for the moment Grey link) to differential category, which hopefully I will be able to add a sutb for later.

• Found this abandoned stub entry from 2010. Have removed the line

Zoran: This is just a reminder for me to work on this entry in few days (to do list)

and instead added some minimum structure, including formatting, a line in the Idea-section (but not doing it justice) and some more references.

• some minimum, for the moment just for the convenience that the link works

(in creating this entry I noticed that we have an ancient stub entry mapping complex that deserves some attention)

• 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?

• The link for ’equivalent’ at the top redirected to natural isomorphism which (as I understand it) is the correct 1-categorical version of an equivalence of functors, but this initially lead me to believe that a functor was monadic iff it was naturally isomorphic to a forgetful functor from the Eilenberg-Moore category of a monad on its codomain, which would mean that the domain of the functor was literally the Eilenberg-Moore category of some adjunction since natural isomorphism is only defined for parallel functors.

• I have added to monoidal model category statement and proof (here) of the basic statement:

Let $(\mathcal{C}, \otimes)$ be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$. If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then 2) the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

$\gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.$

The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

• Fixed a faulty link.

• There’s a first stab at it.

• Created stub.

• I started an idea section at transgression, but it could probably use some going over by an expert. I hope I didn’t mess things up too badly. I was reading Urs’ note on “integration without integration” on the train ride home and fooled myself into thinking I understood something.

By the way, this reminded me of a discussion we had a while back

Integrals: Loops space vs target space

• added an older reference