nLab essentially algebraic (infinity,1)-theory

Definition

Definition

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

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

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

Properties

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

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

Examples

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

Related entries

• algebraic theory, essentially algebraic theory

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

References

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

• Jacob Lurie, Structured Spaces