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Computads for Generalised Signatures

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Peer-reviewed

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Type

Article

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Abstract

We introduce a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe how computads give rise to signatures. Motivated by work of Batanin, we show that computads with certain generator-preserving morphisms form a presheaf category, and describe a forgetful functor from algebras to computads. Algebras free on a computad turn out to be the cofibrant objects for a certain cofibrantly generated weak factorisation system, and the adjunction above induces the universal cofibrant replacement, in the sense of Garner, for this weak factorisation system. Finally, we conclude by explaining how many-sorted structures, weak ω-categories, and algebraic semi-simplicial Kan complexes are algebras of such signatures, and we propose a notion of weak multiple category.

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Keywords

4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences

Journal Title

Journal of Pure and Applied Algebra

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Journal ISSN

0022-4049
1873-1376

Volume Title

Publisher

Elsevier