On left and right model categories and left and right Bousfield localizations

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Motivic slices and coloured operads (2010, background): “c′X X C c′X[1] is a map of exact triangles.”
Model structures on commutative monoids in general model categories (2014, methodology): “In order to make the Reedy structure into a monoidal model category we require two results of Barwick [1].”
A universal characterization of higher algebraic K-theory (2010, methodology): “In model categories, there is a well-developed theory of Bousfield localization (see for example Hirschhorn’s comprehensive discussion in [39], Goerss and Jardine’s treatment in the simplicial setting [34], or the exposition of Smith’s theory for combinatorial model categories in [3]).”
Higher Segal Spaces (2012, unknown): “Enriched Bousfield localization theory, developed in [Bar10], starts with the following definition.”