two rather different capabilities:

1.  Combining vastly different dynamical models into a multi-model system in a systematic way, and

2.  Analyzing homological invariants to study locally- and globally-consistent states of the system.

Both of these leverage the topological structure already inherent in and among the models. For instance:

1.  The base topological space is often essentially a material volume, envisioned abstractly. It can be partitioned into a cellular space, such as finite element meshers already do. All modern solid modelers store an explicit, topological model of the model volume [@Hoffmann_1989 Chap. 2]. In building sheaves, the topology can be refined (cells subdivided, for instance) in order to construct discretizations. Some solid modeling/meshing APIs can do this natively [@Li_2005 Sec. 3].

2.  The local data represented in a sheaf are the state variables in the interior of each mesher cell, exactly as the finite element solver represents them.

3.  The equations are encapsulated in (not necessarily linear) maps deriving boundary values from the parameters known about the interiors of each cell.

Modeling systems with diagrams {#sec:diagram}
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Dynamical models usually involve a collection of state variables and equations that determine their admissible values. For instance, the famous Lorenz system specifies the 