The formalism of an “abelian category” is meant to axiomatize the operations of linear algebra. From there, the notion of “derived category” as the category of complexes “upto quasi-isomorphisms” is natural, motivated in part by topology. The formalism of t-structures allows one to construct new abelian categories which are quite useful in practice (giving rise to new cohomology theories like intersection cohomology, for example). In this talk we want to discuss a notion of punctual (=”point-wise”) gluing of t-structures which is possible in the context of algebraic geometry. The essence of the construction is classical and well known, but the new language leads to useful constructions in the motivic world.