  In this paper, we introduce the concept of 3-alterfolds with embedded
separating surfaces. When the separating surface is decorated by a spherical
fusion category, we obtain quantum invariants of 3-alterfold, which is
consistent with many topological moves. These moves provide evaluation
algorithms for various presentations of 3-alterfold, e.g. Heegaard splittings,
triangulations, link surgeries. In particular, we obtain quantum invariants of
3-manifolds containing surfaces, generalizing those of 3-manifolds containing
framed links. Moreover, in this framework, we topologize fundamental algebraic
concepts. For instance, we implement the Drinfeld center by tube diagrams as a
blow up of framed links. The topologized center leads to a quick proof of the
equality between the Reshetikhin-Turaev invariants and the Turaev-Viro
invariants for spherical fusion categories. In addition, we topologize
half-braiding, $S$-matrix and the generalized Frobenius-Schur indicators, etc.
In particular, the latter leads to a quick topological proof of the
equivariance of the generalized Frobenius-Schur indicators.
