On different ways of being small

Remy Tuyeras – 26 June 2013, 7 August 2013

Since its original statement in 'Homotopical Algebra' (Quillen, 1967), the small object argument has been generalised in several ways. Quite often, its improvement cares about the question of being small in the category where it is applied. An easy way to start with is to suppose that the category is locally presentable. Then, rather than considering a global assumption on the structure of the category, it is possible to generalise the argument by only looking at a certain collection of 'small' objects. But again, the condition of being small may be generalised by considering various sets of parameters providing the objects with various properties. In this talk, I will look at two notions of smallness, the original one, used by Quillen in his book, and a slight modification of it whose particular property is to be 'more adapted' to the study of homotopy (co)limits.