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• 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.

(??)

• Made a remark, to fill in a gap in the constructive proof that group monomorphisms are regular.

• Added a stub of an Idea section.

• Added a description of slant products in cohomology. Added references to Dold’s book.

• expanded copower:

added an Idea-section, an Example-section, and a paragraph on copowers in higher category theory.

• Expanded and reorganised the entry.

• Update diagrams to tikzcd syntax. Also removed comment about using “Leibniz order” for composition of morphisms, which IMHO was unnecessary and confusing.

• am starting something here

• A stub here. Is there a ready example?

• Added reference to the conjectured higher topological topos

• suddenly I found a bunch of further references on this, so I am giving this its own entry now, for ease of recording stuff

• am giving this elementary-but-important fact its own entry, for ease of referencing

• Created page to record some definitions. I am unable to see the relationship between Bousfield’s definition and Joyal’s, although I included some partial results.

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

• some minimum, just so to have a canonical place for linking references jointly from LHC and flavour anomaly

• bare minimum, for the moment just so that one may link to it

• The page split coequalizer said that the canonical presentation of an Eilenberg–Moore algebra is a split coequalizer in the category of algebras. I don’t think that’s right – if I recall correctly it’s reflexive there, but in general not split until you forget down to the underlying category. So I changed the page.

• created a page for Peter Symonds (Manchester)

• Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the permutation representation in the standard representation theory books like in

• Claudio Procesi, Lie groups, an approach through invariants and representations, Universitext, Springer 2006, gBooks

I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.

Edit: new (related) entries for Claudio Procesi and Arun Ram.

• I want to make a list with (historical) null results in experimental physics that have been important for development of theoretical physics. Puny start so far, hope to collect more

• I added a discussion of space in Kant’s Transcendental Aesthetics in Critique of Pure Reason.

By the way, the translation of the quote from Kant in the section “On Aristotelian logic” seem a bit strange: I think the original German sentence was “Begriffe aber beziehen sich als Prädikate möglicher Urtheile auf irgend einen noch unbestimmten Gegenstand” (“But conceptions, as predicates of possible judgements, relate to some representation of a yet undetermined object.”).

PS The automatic function to create this thread in the nforum did not word.