The n-Category Café

Recent advances show that category theory can be used to capture the essential behavioural properties of many complex systems, and provides the right language to study their foundational concepts across a broad range of disciplines, including the physical world, logical and deductive systems, the way that meaning is encoded into a sentence, and closely related, cognition.

I wonder if it might become less plausible as one expands into the “general context of quantum physics”, as you suggest, that there are to be found common structures between the physical world and cognition.

Unless, perhaps, one takes physics to capture not the world as it is in itself, but our cognitive dealings with it. We had a discussion about such matters in the early days of the Café.

Also, I meant to see if I could make some sense of Liang Kong’s remark

…our physical space is nothing but a network of structured stacks of information, from which spacetime can emerge.

mentioned here.

I wonder if it might become less plausible as one expands into the “general context of quantum physics”, as you suggest, that there are to be found common structures between the physical world and cognition.

By the way, in case you or other readers here haven’t seen it yet: that part of the proposal probably arises as an extrapolation of this work:

Bob Coecke, Mehrnoosh Sadrzadeh, Stephen Clark, Mathematical Foundations for a Compositional Distributional Model of Meaning (arXiv:1003.4394)

Abstract We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are “lifted” to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the inner-product can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our `categorical model’ which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montague-style Boolean-valued semantics.

A New Scientist article on this stuff is here.

The first sentence of the second paragraph clarifies, I think: “Essential mathematical components of this work include monoidal categories, sheaf theory and coalgebra.”

All of these are used heavily in computer science as abstractions for modelling dynamical systems.

1. Monoidal categories give models of linear and other substructural logics, which have proof-theoretic readings in terms of state change.

2. Sheaves arise as categorifications of Kripke models. By considering the internal set theory of sheaf categories, we can extend traditional Kripke models to talk about (say) variable sets, and not merely variable propositions.

3. Coalgebra let us model systems in terms of their observable behavior – a class of behaviors is described by the final coalgebra for a functor $F$, and a particular system is a (non-final) coalgebra for this functor $c : A \to F(A)$. We can equate two systems $c : A \to F(A)$ and $d : B \to F(B)$ just when they are bisimilar.

Modal logic seems to provide ideas for potential glue between 2 and 3, as we discussed here.

Eugene and I discussed before coalgebra and dynamical systems.

Hmm, I wonder if there was something worth exploring in that post and first comment.