## 1 Introduction

How should we conceive of concepts, of cognition, or of circuit computation? We argue: compositionally. Composition is the tool by which we construct complex concepts: constantly, informally, and automatically. But it is not just our concepts that are compositional: it has long been noted that our cognitive architecture is modular (fodor1983modularity; marcus2018deep), and much of cognitive neuroscience relies on the well-tested assumption that this modularity maps onto the structure of the brain. Mathematics itself is increasingly recognized as compositional (kock2006synthetic; bauer2017hott), and strong compositionality is increasingly used in software engineering to improve correctness and code reusability (fischer2017haskell; haftmann2010higher)

. Despite this simplifying power, few models in cognitive science and machine learning are actually compositional, even when modelling compositionality itself (

*eg.*, (chang2016compositional; piantadosi2016logical)).

We propose instead taking compositionality seriously, using the mathematics of
composition—category theory—and show how doing so allows us to translate
concepts between contexts and levels: from abstract concepts themselves
(phillips2010categorial; bolt2019interacting) to their possible realization
in a circuit model. This paper is an abridged version of a work in progress,
provisionally to appear in *Compositionality*, and we defer many formal
details and proofs to that manuscript. In §2 we introduce the
background mathematics. In §LABEL:sec:concepts, we introduce the conceptual
setting. §LABEL:sec:circ shows how to translate concepts to circuits, and
§LABEL:sec:back suggests how to translate back again.

## 2 Category theory and the ‘structure of structure’

A category is a very simple structure, capturing only what is necessary to enforce compositionality:

###### Definition 1.

A category is a set of objects such that for any two objects there is a set of arrows obeying a composition rule: for any arrow and any arrow , there is a composite arrow . Every object has an identity arrow, .

Being so general, almost all concepts can be formalized categorically. For example, there is a category whose objects are parts of speech and whose arrows are grammatical relations; and a category of vector spaces and linear maps. Category theory can also be applied to itself: there is a category

of categories, whose arrows are called functors:
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