An approach that interprets a syntactic category (presenting a theory or language) as an object-class in a semantic category by means of functors that preserve the syntactic structure required by the theory; models are the functors that realize syntactic operations as semantic structure-preserving maps.

Principle

Encode syntactic formation rules and inference devices as categorical structure, and let semantics be given by structure-preserving functors from the syntactic category into a chosen semantic category so that compositionality and operations commute with interpretation.

Demonstration

For an equational theory presented by operations and equations one forms a small category whose objects are finite contexts and whose morphisms are equivalence classes of terms; a model is a product-preserving functor from that category into the category of sets, which assigns to each sort a set and to each operation a function respecting the equations.

Misapplication

Treating any mapping of syntactic symbols to semantic values as functorial semantics without checking that the mapping preserves the categorical structure (e.g., products, identities, composition) and thus fails to respect equations or compositional rules.

Consequence

When applied correctly, functorial semantics yields compositional, modular models of theories, reduces model constructions to universal categorical operations, and makes transformations between models correspond to natural categorical morphisms.

Reversal

Instead of interpreting syntax as objects in a semantic category, reverse the map and ask which syntactic presentations are induced by given semantic structures; this emphasizes generation of syntax by semantics rather than interpretation of syntax.

Boundary

Applies where syntax can be organized as a category with the structural features to be preserved (finite products, monoidal structure, etc.); it excludes semantic accounts that cannot be expressed via functors (e.g., purely statistical semantics without compositional structure) or cases where no faithful syntactic category is available.

Semantic Tension

Competes with model-theoretic accounts that present semantics as sets of truth-assignments rather than as functors; the tension is between treating models as set-theoretic structures and treating them as structure-preserving mappings from a canonical syntactic object.

Synthesis

Functorial semantics is the categorical prescription: package a theory as a syntactic category and realize its models as functors into a semantic category so that syntactic composition and structural constraints are reflected exactly by functorial image.