# A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces

arXiv: https://arxiv.org/abs/2110.05985 · HTML: https://mlsanity.com/paper/2110.05985/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 32 references that are in the map.
- First posted on arXiv 12 Oct 2021.

## Builds on (most cited first, 9 of 9)

- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [The Mathematics of Text Structure](https://mlsanity.com/paper/1904.03478/) (2019)
- [Interacting Conceptual Spaces I : Grammatical Composition of Concepts](https://mlsanity.com/paper/1703.08314/) (2017)
- [Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability](https://mlsanity.com/paper/2010.07416/) (2020)
- [A Thorough Formalization of Conceptual Spaces](https://mlsanity.com/paper/1706.06366/) (2017)
- [Autonomization of Monoidal Categories](https://mlsanity.com/paper/1411.3827/) (2014)
- [A Compositional Explanation of the Pet Fish Phenomenon](https://mlsanity.com/paper/1509.06594/) (2015)

## Built on by (most cited first, 4 of 4)

- [Towards Compositional Interpretability for XAI](https://mlsanity.com/paper/2406.17583/) (2024)
- [The Conceptual VAE](https://mlsanity.com/paper/2203.11216/) (2022)
- [From Conceptual Spaces to Quantum Concepts: Formalising and Learning Structured Conceptual Models](https://mlsanity.com/paper/2401.08585/) (2023)
- [Formalising and Learning a Quantum Model of Concepts](https://mlsanity.com/paper/2302.14822/) (2023)

## Abstract (arXiv)

We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within G"ardenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these are the most general choice satisfying a criterion due to G"ardenfors and which are well-behaved compositionally. We then generalise these to define the category of log-concave probabilistic channels between convex spaces, which allows one to model fuzzy reasoning with noisy inputs, and provides a novel example of a Markov category.