Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards enriched universal algebra

Jiří Rosický, Giacomo Tendas

Abstract

Following the classical approach of Birkhoff, we suggest an enriched version of enriched universal algebra. Given a suitable base of enrichment $\mathcal V$, we define a language $\mathbb L$ to be a collection of $(X,Y)$-ary function symbols whose arities are taken among the objects of $\mathcal V$. The class of $\mathbb L$-terms is constructed recursively from the symbols of $\mathbb L$, the morphisms in $\mathcal V$, and by incorporating the monoidal structure of $\mathcal V$. Then, $\mathbb L$-structures and interpretations of terms are defined, leading to enriched equational theories. In this framework we characterize algebras for finitary monads on $\mathcal V$ as models of an equational theories.

Towards enriched universal algebra

Abstract

Following the classical approach of Birkhoff, we suggest an enriched version of enriched universal algebra. Given a suitable base of enrichment , we define a language to be a collection of -ary function symbols whose arities are taken among the objects of . The class of -terms is constructed recursively from the symbols of , the morphisms in , and by incorporating the monoidal structure of . Then, -structures and interpretations of terms are defined, leading to enriched equational theories. In this framework we characterize algebras for finitary monads on as models of an equational theories.
Paper Structure (12 sections, 23 theorems, 49 equations)

This paper contains 12 sections, 23 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Background notions
  3. Enrichment
  4. Pretheories, theories, and monads
  5. Languages
  6. Terms
  7. Enriched equational theories
  8. Main results
  9. Elimination of arities
  10. Enriched Birkhoff subcategories
  11. Multi-sorted languages and theories
  12. More on Birkhoff subcategories

Key Result

Proposition 2.3

An identity-on-objects $\mathcal{V}$-functor $\tau\colon\mathcal{V}_\lambda^{\operatorname{op}}\to \mathcal{T}$ is a $\mathcal{V}_\lambda$-theory if and only if for any $Z\in\mathcal{V}_\lambda$ $\mathcal{V}$-naturally in $X\in\mathcal{V}_\lambda^{\operatorname{op}}$; in other words, if $\mathcal{T}(Z,-)$ preserves $\lambda$-small powers of $I$.

Theorems & Definitions (87)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • ...and 77 more