sec-categorical-doctrines
1.2 Doctrines: the structure known in advance
The word doctrine occurs throughout this book because saying that a learner expects “a category” is rarely precise enough. One learner may expect finite products, another a tensor product, another tangent structure, and another only a declared family of observable questions. A doctrine names this ambient package of admissible structure and the maps allowed to preserve it.
A categorical doctrine \(\mathfrak D\) is a specified 2-category \(\mathbf{Doc}_{\mathfrak D}\) together with a forgetful 2-functor
Its objects are categories equipped with the structure declared by \(\mathfrak D\), its 1-cells are the admitted structure-preserving functors, and its 2-cells are the admitted transformations. The doctrine also fixes which of its 1-cells count as equivalences. When the structure is algebraic, \(\mathbf{Doc}_{\mathfrak D}\) is often presented as the 2-category of pseudoalgebras for a 2-monad on \(\mathbf{Cat}\); a sketch gives another common presentation.
Thus a doctrine is not one particular world and is not normally one particular theory. It specifies the kind of structure a theory or world may carry, how that structure is transported, and the resolution at which two such structures are identified.
A \(\mathfrak D\)-theory is a small object \(\mathbb T\) of \(\mathbf{Doc}_{\mathfrak D}\). If \(\mathcal V\) is a semantic object of the same doctrine, a model of \(\mathbb T\) in \(\mathcal V\) is an admitted 1-cell
Consequently the category of models is the hom-category \(\mathbf{Doc}_{\mathfrak D}(\mathbb T,\mathcal V)\).
The hierarchy is easiest to see by comparing mathematical, developmental, and scientific examples.
- Finite-product doctrine.
Its objects are categories with finite products and its maps preserve those products (strictly or coherently, according to the declared convention). A Lawvere theory is a small theory in this doctrine with a distinguished generator. The Lawvere theory of groups is one such theory; a product-preserving functor \(\mathbb T_{\mathrm{Grp}}\to \mathbf{Set}\) is a group. Hence groups do not constitute the doctrine: they are models of a theory living within it.
- Symmetric-monoidal doctrine.
Its objects are symmetric monoidal categories and its maps are admitted symmetric monoidal functors. The category \((\mathbf{Vect}_k,\otimes ,k)\) is an instance of this doctrine, and its objects are vector spaces. A PROP is a particularly useful small theory in the doctrine; a symmetric monoidal functor from that PROP into \(\mathbf{Vect}_k\) realizes its abstract operations as multilinear structure. Associators, unitors, and braidings record the coherence required when tensor products are not treated as literally strict.
- Core-knowledge doctrine.
Spelke’s hypothesis admits a categorical reading as a claim about the kinds of world models for which a child is prepared. The proposed doctrine is heterogeneous: it combines constraints on objects, approximate magnitude, space, form, goal-directed agents, and social-linguistic interaction, together with interfaces among them. A child does not begin knowing which particular objects or people exist, which language is spoken, or which regularities hold locally. Those are theories and models to be identified within the restricted category of possible worlds. Section 4.1 develops this proposal and its empirical burden.
- Symmetry doctrine for matter.
Particle physics treats admissible entities and processes through group actions, representations, equivariant maps, tensorial composition, and invariance laws. At the kinematic level, relativistic elementary particle types are classified by suitable irreducible representations of the Poincaré group; internal gauge symmetries add further representation data [ Wigner , 1939 ] . Schematically, the doctrine says that physical theories and their maps must respect such symmetry and compositional structure. A particular choice of spacetime and gauge groups, matter representations, interaction generators, and relations belongs to a theory within that doctrine. Particle species and scattering processes then appear as particular objects and morphisms in its models. Section 0.8 explains how collider interaction probes this structured hypothesis space.
These examples separate four levels that later repair mechanisms must not confuse:
Changing a group while retaining the group laws changes a model. Changing the equations changes the theory. Abandoning finite products as the governing notion of combination changes the doctrine. Likewise, learning a new word or object need not alter the child’s core doctrine, and discovering a new particle need not abandon symmetry. In either case, doctrinal repair is the stronger move: it changes the prior notion of which worlds and transports are admissible, rather than adding another inhabitant to the current world.
We use two qualified variants of the term. A preservation doctrine specifies which designated limits, colimits, tensors, or other constructions a realization must preserve. A query doctrine specifies an admissible family of probes, its closure rules, and the resulting observational equivalence of hypotheses. A core doctrine may bundle structural, preservation, and query commitments into the learner’s prior. These are deliberate extensions of the same organizing idea: each states what structure is typed, what maps respect it, and what comparisons count.
“Doctrine” is not a decorative synonym for “assumption.” Every doctrine used below must expose at least its structured objects, admitted maps, and equivalences. If a theorem also depends on chosen probes or preservation requirements, those must be declared separately. This is what makes assimilation within a doctrine distinguishable from accommodation that repairs the doctrine itself.