I would like to know the most commonly used terminology for the following simple object in combinatorics/topology. I restrict to the two dimensional case since this is what I am mainly interested in:
?-complex. Take a collection of finite polygons in $\mathbb{R}^{2}$, then given a pairing of the set of all edges of the polygons, and an orientation for each edge, we glue along this pairing respecting the orientation) to form a certain topological space.
It seems that the notion of polyhedral complex is nearly what I am looking for, but the definitions all assume that the polygons are embedded in the same Euclidean space. I don't want to make such an assumption. Or maybe there is a theorem that say that such an object can always be given the structure of polyhedral complex? although that seems a bit unneccesary, I am mainly hoping the above object appears in some book.
Yes, there is an abstract definition in the literature and one allows higher-dimensional polytopes, not just planar polygons:
Definition A.2.12 in
Michael W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008.
See also Chapters I.7 and II.5 in the book
M. Bridson and A. Haefliger and "Metric Spaces of Non-Positive Curvature," Springer-Verlag, 1999.
As for an embedding the a Euclidean space, this is quite false. For instance, take two equilateral triangles glue all their edges in pairs via isometries. The result has three vertices, three edges and two 2-dim faces.