We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.

We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences.

We show that the subcategory of mixed Tate motives in Voevodsky’s derived category of motives is not closed under infinite products. In fact, the infinite product $\prod _{n=1}^{\infty }\mathbf{Q}(0)$ is not mixed Tate. More generally, the inclusions of several subcategories of motives do not have left or right adjoints. The proofs use the failure of finite generation for Chow groups in various contexts. In the positive direction, we show that for any scheme of finite type over a field whose motive is mixed Tate, the Chow groups are finitely generated.

We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.