sec-uocl-limits-colimits

6.8 Limits, colimits, and multimodal fusion

The symmetric monoidal product places learners side by side but imposes no agreement. Genuine multimodal fusion begins when local hypotheses are observed in a shared doctrine. For a visual learner and a linguistic learner, suppose there are comparison functors

\[ O_V:\mathcal H_V\longrightarrow \mathcal K, \qquad O_L:\mathcal H_L\longrightarrow \mathcal K, \]

where \(\mathcal K\) expresses shared claims such as object identity, persistence, agency, or reference. Their compatible joint state is the indexed pullback

\begin{equation} \mathcal H_{V\bowtie L} =\mathcal H_V\times _{\mathcal K}\mathcal H_L. \end{equation}

6.1

Proposition 6.12 Universal multimodal fusion

If the pullback in Equation 6.1 exists in every presentation fiber, is preserved by reindexing, and the two updates preserve the compatibility comparison, then it defines a fused UOCL learner. Every joint learner equipped with compatible maps to the visual and linguistic learners factors uniquely through it, up to the declared equivalence.

Proof ▼

Fiberwise pullbacks provide compatible joint hypotheses and their projection maps. Preservation by reindexing makes these fibers an indexed category; update preservation closes it under online learning steps. The factorization and its uniqueness are exactly the pullback universal property, interpreted fiberwise and then assembled by reindexing coherence.

This universal property distinguishes fusion from concatenating embeddings. It gives a minimal joint learner containing both local hypotheses subject to the declared cross-modal agreements. A contradiction can then be localized: the visual hypothesis, the linguistic hypothesis, or their comparison map may need repair. An undifferentiated joint parameter vector provides no analogous structural diagnosis.

Other universal constructions have different meanings. Products support parallel observation and joint queries. Equalizers enforce agreement between two proposed translations. Coproducts retain tagged alternative learners. Pushouts glue learned worlds along an already identified common subworld, and more general colimits assemble a world from overlapping fragments. None is an automatic operation: existence, effective construction, preservation of warranted answers, and avoidance of spurious identifications must be proved for the selected UOCL doctrine.

The tangent refinement adds one decisive compatibility test. If the tangent functor preserves the pullback used for fusion, then

\[ T(\mathcal H_V\times _{\mathcal K}\mathcal H_L) \simeq T\mathcal H_V\times _{T\mathcal K}T\mathcal H_L. \]

In that case, fusing first and learning infinitesimal variation commute. If the comparison fails, cross-modal perturbations contain information absent from the separate tangent learners; the fusion mechanism itself must be learned or accommodated. This supplies a concrete way in which the category of UOCL learners creates the conditions required by DIAL.