a map stores information about the territory in a smaller amount of space. — Pfhorrest
Yes, a map is a symbolic and simplified representation of a territory, made for a certain purpose e.g. facilitating the analysis of, or navigation within the territory.
So the map is made for a specific objective, a certain type of use, a teleology. It implies a goal, or several goals.
These goals drive the kind of simplification applied when building the map. The features of the territory that have no import for the goal are not represented. For instance, roads maps -- designed to facilitate road travel and transportation -- typically do not represent vegetation cover or elevation.
Without such simplification the map would be useless, or more precisely, it would have no advantage compared to the territory. It would also be impossible to build.
Finally, the map is static while the territory is dynamic. The map represents a state of the territory at time t. There are exceptions to this, eg chronograms that map space-time. But the chronogram itself is drawn at time t.
In the case of man-made maps or the regular kind, the map is given some stable material support e.g. written down on paper. What is written down is a symbolic, mind-derived map, a mental map. All man-made maps are mental maps, originally.
Therefore the relationship between map and territory is not symetrical. The "roles" cannot be reversed. There is an fundamental epistemic cut between them. If we were to explore a new planet, and found a map carved on some stone of the nearby terrain, we would conclude that a conscious being carved it.
Now, where does that lead us re. Dualism vs. Monism?
The mind itself can be seen as a geographer, drawing upon a collection of mental maps, constantly updated. Since there is no form without matter, even mental maps (mental symbolic representations of the world) must be coded onto some physical support.
It stands to reason that they are written down on neurons.
This is a dualist perspective in the sense that mental events are recorded onto neurons (and from neurons onto paper or other material support) for later recalling into the mental world. It's mind over matter.
Now my challenge to you is to make sense of the map-territory relationship in monist language.