© 2022 World Scientific Publishing Company.Fields exhibit a variety of topological properties, like different topological charges, when field space in the continuum is composed by more than one topological sector. Lattice treatments usually encounter difficulties describing those properties. In this work, we show that by augmenting the usual lattice fields to include extra variables describing local topological information (more precisely, regarding homotopy), the topology of the space of fields in the continuum is faithfully reproduced in the lattice. We apply this extended lattice formulation to some simple models with nontrivial topological charges, and we study their properties both analytically and via Monte Carlo simulations.