CT 2007 HLCGB

[pozorvlak]

High: hanging out on the beach after the conference dinner with Tom, Steve Lack, Eugenia Cheng, Peter Lumsdaine, Emily and Louise, and Telyn Kusalik. Great fun, good company, an amazing starfield, and cold but invigorating water to swim in :-) Also, in general, the surroundings: Carvoeiro's a beautiful place

Comments

[pozorvlak]

OK, I'm still not at all clear on semi-abelian categories, but here goes: a category is homological if it has a zero object (initial + terminal), it has finite limits, it's regular (er, not sure what that means), and Bourn protomodular (ie, satisfies the Short Five Lemma, though there's some more abstract definition). It's semi-abelian if, in addition, it's exact and has finite colimits. Semi-abelian categories are strictly more general than abelian categories (a category C is abelian iff both C and C^op are semi-abelian), but they don't have, say, enrichment over Ab: you can't add arrows together like you can in abelian categories. Semi-abelian categories include things like the category of groups, Lie algebras, rings (IIRC), and more generally any Mal'cev variety. Beyond that, I'm not sure: I think you can re-state the "alternating sum" definition of the homology of a simplical complex in some way that doesn't involve taking sums, and hence do the bar-resolution stuff as usual. This (again, I think) leads to the usual definition of group (co)homology. I'm afraid I don't know of any accessible papers giving the basics of this stuff, but I'll ask around. FWIW, the main people working on this seem to be Dominique Bourn, George Janelidze and his son and daughter, Rutge Kieboom, Marino Gran, Francis Borceux, Tim van der Linden and Thomas Everaert, and possibly Walter Tholen, but I've undoubtedly forgotten lots of people.

A model category is a complete and cocomplete category K, equipped with three distinguished classes of morphisms W, F and C such that
1) W has the "2 out of 3" property and is closed under retracts.
2) (W,F) and (W,C) are weak factorization systems.

Unfortunately, I didn't manage to write down the definitions of the 2 out of 3 property or of a weak factorization system :-) But a WFS is essentially defined by taking the standard notion of a factorization system (see eg Borceux's Handbook of Categorical Algebra) and removing the word "unique", as I understand it. So, we haven't simplified the problem all that much, but we have at least broken it down a bit. This paper looks relevant.