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

[michiexile]

Some other thoughts: the semi-abelian guys' project to generalise the whole of homology theory to the semi-abelian setting seems to be proceeding apace, and now that I've learned a bit about abelian categories I finally feel like I've got some sense of what they're trying to do. Either quantales are growing in importance, or they've been important for ages and I've only just noticed - they were certainly much in evidence. Similarly, lots of talks involved model categories, which now seem to be the default way of attacking categories equipped with some notion of "sameness" - obviously model categories are not just for geometers any more... Sounds like I should make some effort to understand those two things: no doubt for the latter I shall be substantially aided by the handy three-line definition of model category given by Jiri Rosicky :-) Similarly, I think I really need to buckle down and learn some categorical universal algebra (sketch theory and so on) - who knows, it might even be useful...

Ooooook, you have me now. I would kindly request you fill me in on everything you know yourself of at least the following bits:
1) I want to know how they generalize homology to semi-abelian categories. I also want to know what a semi-abelian category is and how it relates to the abelian categories.

2) PLEASE give me Jiri's three-liner!

Sounds like you had a blast. I'm getting envious.

[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.