Triangulated Categories Part II

The results in this note are all from [Nee01] or [BN93]. For necessary background on cardinals (particularly regular and singular cardinals) see our notes on Basic Set Theory (BST). In our BST notes there is no mention of grothendieck universes, whereas all our notes on category theory (including this one) are implicitly working inside a fixed grothendieck universe U using the conglomerate convention for U (FCT,Definition 5). So some explanation of how these two situations interact is in order. Definition 1. The expression ordinal has the meaning given in (BST,Definition 2), and a cardinal is a special type of ordinal (BST,Definition 7). Under the conglomerate convention, ordinals and cardinals are a priori conglomerates, not necessarily sets. We say an ordinal α is small if it is a small conglomerate. We make the following observations • If α ≺ β are ordinals with β small, then α is also small. • If an ordinal α is small, so is α = α ∪ {α}. • The first infinite cardinal ω = א0 is always small, by our convention that grothendieck universes are always infinite (FCT,Definition 4). Therefore every finite cardinal is small. Remark 1. Although in our BST notes we consider the finite cardinals 0, 1 to be regular, it is our convention throughout this note that all regular cardinals are infinite. This just saves us from writing expressions like “small infinite regular cardinal” repeatedly. Remark 2. Recall that when we say a category C is complete, or even just has coproducts, we mean that all set-indexed colimits (resp. coproducts) exist in C (we are working under the conglomerate convention, so a set is an element of our universe U). If β is a small cardinal, we say that C has β-coproducts if any family of objects {Xi}i∈I in C indexed by a set I of cardinality < β has a coproduct. For example, C has finite coproducts iff. it has א0-coproducts and has countable coproducts iff. it has א1-coproducts.