In a previous paper, we studied the indeterminacy of the value of a derived functor at an object using different definitions of a derived functor and different types of fibrant replacement. In the present work we focus on derived or homotopy limits, which of course depend on the model structure of the diagram category under consideration. The latter is not necessarily unique, which is an additional source of indeterminacy. In the case of homotopy pullbacks, we introduce the concept of full homotopy pullback by identifying the homotopy pullbacks associated with three different model structures of the category of cospan diagrams, thus increasing the number of canonical representatives. Finally, we define generalized representatives or models of homotopy limits and full homotopy pullbacks. The concept of model is a unifying approach that includes the homotopy pullback used by J. Lurie and the homotopy fiber square defined by P. Hirschorn in right proper model categories. Properties of the latter are generalized to models in any model category.
Keywords: Localization, model category, homotopy category, derived functor, homotopy limit, homotopy pullback, homotopy fiber square
2020 MSC: 18E35, 18N40, 14A30
Theory and Applications of Categories, Vol. 38, 2022, No. 41, pp 1608-1635.