## ❶ General background

- Human language grammar works by
**types**(e.g., noun, verb, tense). - The grammatical type inventory, abstractly speaking, is a
**structured set**. - This set is structured at multiple
**levels of abstraction**.

## ❷ Particular background

- At a certain level of abstraction, grammatical types are structured in
**posets**(called "extended projections"). - Each extended projection is defined by a
**major part of speech**, which encompasses types of the same part of speech (e.g., tense and aspect are both**verbal**types). - There is an intuitive
**parallelism**across extended projections (e.g., tenses anchor verbal events in time \(\approx\) demonstratives anchor nominal entities in space). - Extended projections vary in size and content but the parallelism is
**always**there. - The parallelism is not strictly one-to-one but more exactly
**zone-to-zone**. - Two major ways to formulate the zones:
- \(\{\)core\(\}\) \(\rightarrow\) \(\{\)inflection\(\}\) \(\rightarrow\) \(\{\)discourse\(\}\)
- \(\{\)classification\(\}\) \(\rightarrow\) \(\{\)point-of-view\(\}\) \(\rightarrow\) \(\{\)anchoring\(\}\) \(\rightarrow\) \(\{\)linking\(\}\)

My abstract adopts the second but the results are not contingent on the formulation.

## ❸ Examples

A deliberately complex example:

*Incomparable elements are subtle variants (aka "flavors") of the same type.*

A simpler example:

*This language does not have grammaticalized tense but have classifiers.*

An even simpler example:

*These two extended projections have a very coarse granularity level (i.e., the linguist who uses them is glossing over a lot of details).*

## ❹ Category theory

An extended projection can be viewed as a **poset category**.

- Objects: grammatical types
- Morphisms: instances of the partial order

**Functors** (i.e., monotone functions) can model cross-extended-projection connections.

**Question: What is the parallelism in categorical terms?**

## ❺ Category comparison

The parallelism reflects **similarity** between extended projections. Three **levels of similarity** between categories:

~~Isomorphism (strong)~~~~Equivalence (middle)~~- Adjoint situation (weak)

1 and 2 are ruled out

- extended projections have varied sizes
- poset categories are equiv
**iff**they are iso

*So the parallemism is at best an adjunction if it can be categorically modeled at all.*

## ❻ Mediated adjoint situation

Unfortunately **no direct adjunction** between extended projections is linguistically meaningful (see my dissertation for detail).

*In general many valid mathematical configurations are linguistically counterintuitive (and hence inapplicable).*

** Solution:** there

**is**a linguistically meaningful adjunction between any extended projection poset (\(\mathbf{EP}_i\)) and the

**zone chain**(\(\mathbf{EP}_0\)).

The **left adjoint** \(F_i\) sends grammatical types to their zones, and the **right adjoint** \(G_i\) sends each zone to a "representative" type.

This representative cannot be randomly chosen. The linguistically meaningful choice is the type at the zone's **upper boundary**.

*...because the "meaning" of each zone is the completion of a subderivation (e.g., the inflectional part of a sentence).*

## ❼ Freyd's adjoint functor theorem

Category theory imposes a **constraint** on the above-mentioned adjoint situation:

**Theorem:**Left adjoints preserve colimits (i.e., joins in the case of posets).

**Corollary 1:**grammatical types sitting at zone upper boundaries

**cannot**have incomparable variants (i.e., they cannot be "flavored").

*...because if they do there will be no more adjunction (and hence no more parallelism).*

## ❽ Uniqueness of phase heads

If our category-theoretic modeling is on the right track, it has a nontrivial consequence for **phase theory** (a key component of current Chomsky-school generative syntax):

- Phases are relatively complete subderivations ready for phonological/semantic interpretation.
- Phase-demarcating types (aka "phase heads") are a subset of the zone upper boundary types.

**Corollary 2:**no phase heads can be flavored.

In particular, this makes the "flavored little v heads" (e.g., v_{do}, v_{agent}) questionable on an ontological level (assuming v is a phase head).

Some linguists already argued against them (see my abstract for detail). Category theory provides a rigorous mathematical argument.