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essentially algebraic (infinity,1)-theory

Definition

Definition

An (finitary) essentially algebraic $(\mathrm{\infty },1)$-theory is an (∞,1)-category $T$ with (finite) (∞,1)-limits.

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

$$A:T\to 𝒳.$$A : T \to \mathcal{X} \,.

Properties

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

Examples

• A geometry (for structured (∞,1)-toposes) is an essentially algebraic $(\mathrm{\infty },1)$-theory.

Related entries

• algebraic theory, essentially algebraic theory

• algebraic (∞,1)-theory, essentially algebraic $(\mathrm{\infty },1)$-theory

References

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

• Jacob Lurie, Structured Spaces