Posets arising from decompositions of objects in a monoidal category

Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite height, we define decompositions and partial decompositions of $X$ which are coherent with $\sqcup$, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness. In well-studied scenarios, the poset of $\sqcup$-complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of non-degenerate subspaces of a vector space with a non-degenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.