Presenting Neural Networks via Coherent Functors
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917413893701632 |
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| author | Pugh, Matthew Grundy, Jo Cirstea, Corina Harris, Nick |
| author_facet | Pugh, Matthew Grundy, Jo Cirstea, Corina Harris, Nick |
| contents | This paper develops a methodology for representing machine learning models as models of formal theories, grounded in the perspective that machine learning models are a form of database and that databases are models of theories in coherent logic. Two intermediate results support this approach: any functorial database schema has an associated $κ$-coherent theory whose models coincide with its instances, and data may be hard-coded into a coherent category such that any model of the resulting theory necessarily contains it. These tools are used to show that any dense feed-forward neural network architecture over the floating point numbers may be presented as a coherent category $G$ whose $Set$-models are the networks of that architecture, with inference arising as the precomposition functor $Coh(ι, Set)$ along a coherent functor $ι: RSpan(a_0, a_n) \rightarrow G$. This representation is extended to networks with weight and bias fixing and tying, encompassing sparse and convolutional architectures, via a 2-coequaliser construction in $Coh_\sim$. Taken together, these results recast neural network inference as an extension problem in the 2-category $Coh_\sim$ of coherent categories, supporting the interpretation of a network architecture as a formal hypothesis about the structure of data and of model training as a lifting of a dataset into a more constrained theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Presenting Neural Networks via Coherent Functors Pugh, Matthew Grundy, Jo Cirstea, Corina Harris, Nick Category Theory This paper develops a methodology for representing machine learning models as models of formal theories, grounded in the perspective that machine learning models are a form of database and that databases are models of theories in coherent logic. Two intermediate results support this approach: any functorial database schema has an associated $κ$-coherent theory whose models coincide with its instances, and data may be hard-coded into a coherent category such that any model of the resulting theory necessarily contains it. These tools are used to show that any dense feed-forward neural network architecture over the floating point numbers may be presented as a coherent category $G$ whose $Set$-models are the networks of that architecture, with inference arising as the precomposition functor $Coh(ι, Set)$ along a coherent functor $ι: RSpan(a_0, a_n) \rightarrow G$. This representation is extended to networks with weight and bias fixing and tying, encompassing sparse and convolutional architectures, via a 2-coequaliser construction in $Coh_\sim$. Taken together, these results recast neural network inference as an extension problem in the 2-category $Coh_\sim$ of coherent categories, supporting the interpretation of a network architecture as a formal hypothesis about the structure of data and of model training as a lifting of a dataset into a more constrained theory. |
| title | Presenting Neural Networks via Coherent Functors |
| topic | Category Theory |
| url | https://arxiv.org/abs/2604.15100 |