nLab
enriched Lawvere theory

Context

Higher algebra

Algebraic theories

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Algebras and modules

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Higher algebras

  • symmetric monoidal (∞,1)-category of spectra

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Model category presentations

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Geometry on formal duals of algebras

Theorems

Contents

Idea

The notion of enriched Lawvere theory is a generalization of Lawvere theories to the setting of enriched categories.

Definitions and elementary properties

Definition

An enriched Lawvere theory over a locally finitely presentable? VV-enriched category AA is a small VV-enriched category LL together with a VV-enriched functor J:A f opLJ\colon A_f^{op}\to L that induces an identity map on the set of objects. Here A fA_f denotes the full subcategory of finitely presentable objects in AA. The category VV is a locally finitely presentable closed symmetric monoidal category.

Definition

An algebra over an enriched Lawvere theory is an object XX in AA equipped with a functor M:LVM\colon L\to V whose precomposition with JJ is isomorphic to the representable functor of XX.

Proposition

For the case A=SetA=Set the above definition recovers the usual notion of a Lawvere theory and an algebra over a Lawvere theory.

Proposition

The forgetful functor from algebras to AA is a finitary enriched monadic functor? (over VV).

Proposition

The category of enriched Lawvere theories over AA is equivalent to the category of finitary enriched monads? over AA. The corresponding VV-enriched categories of algebras are also equivalent.

References

Last revised on June 5, 2018 at 03:52:14. See the history of this page for a list of all contributions to it.