Todd,

when you see this here and have a minute, would you mind having a look at *monoidal category* to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?

Thanks!

]]>I always found that the formulation of the theory of infinity-algebras over an (infinity,1)-operad in Lurie’s *Higher Algebra* nicely captures what Baez-Dolan called the *microcosm principle*.

I have added a corresponding remark to the latter entry and expanded slightly.

]]>following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

]]>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.

]]>I have tried to give *algebraic topology* a better Idea-section.

On occasion of Alexander Schenkel’s most recent talk (here) I am finally splitting off an entry *homotopical algebraic quantum field theory* from *AQFT*.

gave *Bousfield localization of spectra* a more informative Idea-section

started a stub for *ambidextrous adjunction*, but not much there yet

created *red-shift conjecture*

*sylleptic monoidal 2-category*, *symmetric monoidal 2-category*

… but now I am running out of steam…

]]>have added to (infinity,1)-operad the basics for the “$(\infty,1)$-category of operators”-style definition

]]>at *braided monoidal 2-categiry* the following query box was sitting, which I hereby move from there to here

+–{: .query} Ben Webster: I would very much like to know: what structure on a (triangulated/dg-/stable infinity/whatever you like) monoidal category would make its 2-category of module categories (give that phrase any sensible construal you like) is braided monoidal.

If one decategorifies this question, one gets the question “what structure on a ring makes its category of representations braided monoidal” and the answer to this question is well-known: a quasi-triangular quasi-Hopf structure.

I asked a MathOverflow question on the same topic. No interesting answers yet. =–

]]>added a tiny bit of basics to *complex oriented cohomology theory*

the entry *modular tensor category* was lacking (among many things that it is still lacking) some pointers to literature that reviews the relation to QFT. I have added a handful, maybe the best one is this here:

- Eric Rowell,
*From quantum groups to Unitary modular tensor categories*, Contemporary Mathematics 2005 (arXiv:math/0503226)

The MO question *Where do all the projection formulas come from?* made me split off a dedicated entry *projection formula*. Also redirecting *reciprocity*.

added to *polynomial monad* the article by Batanin-Berger on homotopy theory of algebras over polynomial monads.

I have edited at *HQFT*, touched the general formatting and structuring a good bit, trying to clean it up and beautify it a bit, and added a brief cross-pointer to the cobordisms hypothesis for cobordisms with maps into a base manifold.

created an entry *twisted Umkehr map*. The material now has some overlap with what I just put into *Pontrjagin-Thom collapse map*. But that doesn’t hurt, I think.

created *cobordism theory determining homology theory* with the basic references (any more results along these lines?), also added a brief cross-link paragraph at *Landweber exact functor theorem*.

I couldn’t think of a better title, suggestions are welcome.

]]>stub for *Landweber exact functor theorem*, to be expanded

The Idea-section at *quasi-Hopf algebra* had been confused and wrong. I have removed it and written a new one.

I have type the definition of multiplicative unreduced generalized cohomology theories $A$ into *multiplicative cohomology theory*. Then I added the statements and their arguments (here) for the compatible $A^\bullet(\ast)$-module structure on $A$-cohomology groups and the $A^\bullet(\ast)$-linearity of the differentials in any Atiyah-Hirzebruch spectral sequence with coefficients in $A$.

After scanning a bunch of literature, my favorite survey of the Adams spectral sequence is now this gem here:

- Dylan Wilson
*Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra*, lecture at 2013 Pre-Talbot Seminar (pdf)

started *K3-spectrum* (K3-cohomology)

added it to the *chromatic tower examples - table*