Abstract We provide a toolbox of extension, gluing, and assembly techniques for factorization algebras. Using these tools, we fill various gaps in the literature on factorization algebras on stratified manifolds, the main one being that constructible factorization algebras form a sheaf of symmetric monoidal ‐categories. Additionally, we explain how to assemble constructible factorization algebras from the data on the individual strata together with module structures associated to the relative links; thus answering a question by Ayala. Along the way, we give detailed proofs of the following facts which are also of independent interest: constructibility is a local condition; the ‐category of disks is a localization of any sufficiently fine poset of disks; constructibility implies the Weiss condition on disks; constructible factorization algebras are algebras for the ‐operad of embedded disks. For each of these, variants or special cases already existed, but they were either incomplete or not general enough.
Karlsson et al. (Sun,) studied this question.