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Fibrant in what model structure? Simplicial abelian sheaves (and presheaves) are fibrant in the projective model structure because all simplicial abelian groups are fibrant. Simplicial abelian sheav …

Taking stalks always commutes with taking homotopy orbits, since filtered colimits of simplicial sets are also filtered homotopy colimits, and homotopy colimits commute with homotopy colimits. Taking …

To answer the question as it is stated: $U$ is an object in a locally presentable category, therefore $U$ is a small object, hence the corepresentable functor of $U$ preserves $α$-filtered colimits fo …

There are two ways to make this construction work. The first way is to iterate the step $F↦F^†$ transfinitely many times. The reason that a single iteration of $F↦F^†$ is not sufficient is that while …