nLab essentially algebraic (infinity,1)-theory

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Definition

Definition

An (finitary) essentially algebraic (,1)(\infty,1)-theory is an (∞,1)-category TT with (finite) (∞,1)-limits.

An algebra over an essentially algebraic (,1)(\infty,1)-theory in some (∞,1)-topos 𝒳\mathcal{X} is a (finite) (,1)(\infty,1)-limit preserving (∞,1)-functor

A:T𝒳. A : T \to \mathcal{X} \,.

Properties

  • An ordinary essentially algebraic theory is a 0-truncated essentially algebraic (,1)(\infty,1)-theory.

  • If XX is an (∞,1)-topos, TT-algebras in 𝒳\mathcal{X} correspond to left exact left adjoints PSh(T)𝒳PSh(T) \to \mathcal{X} under the equivalence Fun L(PSh(T),𝒳)Fun(T,𝒳)\Fun^L(PSh(T), \mathcal{X}) \to \Fun(T, \mathcal{X})

Examples

References

Algebras over essentially algebraic (,1)(\infty,1)-theories that play the role of structure sheaves of algebras are considered in

Last revised on March 12, 2024 at 17:48:59. See the history of this page for a list of all contributions to it.