# Free compact 2-categories

@article{Preller2007FreeC2, title={Free compact 2-categories}, author={Anne Preller and Joachim Lambek}, journal={Mathematical Structures in Computer Science}, year={2007}, volume={17}, pages={309 - 340} }

Before one can attach a meaning to a sentence, one must distinguish different ways of parsing it. When analysing a language with pregroup grammars, we are thus led to replace the free pregroup by a free compact strict monoidal category. Since a strict monoidal category is a 2-category with one 0-cell, we investigate the free compact 2-category generated by a given category, and describe its 2-cells as labelled transition systems. In particular, we obtain a decision procedure for the equality of…

## Topics from this paper

## 53 Citations

Compact Monoidal Categories from Linguistics to Physics

- Physics
- 2010

This is largely an expository paper, revisiting some ideas about compact 2-categories, in which each 1-cell has both a left and a right adjoint. In the special case with only one 0-cell (where the…

Toward discourse representation via pregroup grammars

- Computer Science, MathematicsJ. Log. Lang. Inf.
- 2007

Every pregroup grammar is shown to be strongly equivalent to one which uses basic types and left and right adjoints of basic types only. Therefore, a semantical interpretation is independent of the…

Toward Discourse Representation via Pregroup

- Mathematics
- 2007

Every pregroup grammar is shown to be strongly equivalent to one which uses basic types and left and right adjoints of basic types only. Therefore a semantical interpretation is independent of the…

Complexity of Grammar Induction for Quantum Types

- Mathematics, Computer ScienceQPL
- 2014

It is shown that for a variety of type systems, including pivotal and compact closed categories, the grammar induction problem is NP-complete, and could be extended to linguistic type systems such as autonomous or bi-closed categories.

Pregroup Grammars and Context-free Grammars

- 2007

Pregroup grammars were introduced by Lambek [20] as a new formalism of type-logical grammars. They are weakly equivalent to context-free grammars [8]. The proof in one direction uses the fact that…

Agreement and Cliticization in Italian: A Pregroup Analysis

- Computer ScienceLATA
- 2010

This paper presents an analysis of features specification and agreement based on the parallel computations of a type calculus involving two pregroups grammars: the free pregroup of syntactic types…

Cyclic pregroups and natural language: a computational algebraic analysis

- Computer ScienceCILC
- 2011

This paper proposes an extension of the calculus of pregroups by introducing appropriate cyclic rules that will allow the grammar to formally analyze and compute word order and movement phenomena in different languages such as Persian, French, Italian, Dutch and Hungarian.

Clitic Movement in Pregroup Grammar: A Cross-Linguistic Approach

- Computer Science, MathematicsTbiLLC
- 2009

This paper proposes an extension of the calculus of pregroups including two cyclic rules and uses them to formally analyze movement of clitic clusters in Persian, French, and Italian.

Autonomization of Monoidal Categories

- Mathematics, Computer ScienceACT
- 2019

It is shown that contrary to common belief in the DisCoCat community, a monoidal category is all that is needed to define a categorical compositional model of natural language, and the applications of this principle to various distributional models of meaning are illustrated.

Towards functorial language-games

- Computer Science, MathematicsCAPNS@QI
- 2018

This work compositionally build game-theoretic semantics of sentences by taking the semantic category to be the category whose morphisms are open games, which requires some modifications to the grammar category to compensate for the failure of open games to form a compact closed category.

## References

SHOWING 1-10 OF 32 REFERENCES

Type Grammar Revisited

- Mathematics, Computer ScienceLACL
- 1997

This work explores the assignment of elements of a free protogroup to English words as an aid for checking which strings of words are well-formed sentences, though ultimately the requirement of freeness may have to relax.

Coherence for compact closed categories

- Mathematics
- 1980

It gives us great pleasure to honour, on the occasion of his seventieth birthday, our friend and mentor Saunders MacLane; and to acknowledge that our interest in coherence problems stems from his…

Efficiency of Pregroups and the French Noun Phrase

- Computer ScienceJ. Log. Lang. Inf.
- 2005

An algorithm of complexity n3 to solve the reduction problem in an arbitrary free pregroup as well as recognition by a pregroup grammar is presented and specified to run in linear time.

Nonsymmetric *-Autonomous Categories

- Computer Science, MathematicsTheor. Comput. Sci.
- 1995

This paper provides the definition of ∗ -autonomous categories and develops some equivalent forms of the definition and gives a nonsymmetric version of the Chu construction, which has proven so useful in the past.

Automata, languages, and machines. A

- Computer SciencePure and applied mathematics
- 1974

This book attempts to provide a comprehensive textbook for undergraduate and postgraduate mathematicians with an interest in formal languages and automata, written by Professor Ian Chiswell.

Trends in Logic

- Computer Science
- 1989

This chapter describes the dynamics of model theory and some threats of set theory, which provides universe and semantics more relevant than the set-theoretic ones, and the topological structure of definable sets.

Cut Elimination in Categories

- Mathematics
- 1999

Preface. Introduction. 1. Categories. 2. Functors. 3. Natural Transformations. 4. Adjunctions. 5. Comonads. 6. Cartesian Categories. Conclusion. References. Index.

Bicategories in algebra and linguistics

- Artificial Intelligence
- 2004

in: A

- Lecomte et al., (eds), Logical Aspects of Computational Linguistics, Springer, LNAI 1582, pp. 1-27,
- 1999

Cut elimination for the Lambek calculus of adjoints

- 2002