higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
The notion of a topos $X$ that is equipped with a local algebra-object $\mathcal{O}_X$ is a generalization of the notion of a locally ringed topos. The algebra object $\mathcal{O}_X$ is then also called the structure sheaf.
For that reason in ([Lurie]) such pairs $(X, \mathcal{O}_X)$ are called structured toposes. But since the notion of locally ringed topos is a special case, maybe a more systematic and descriptive term is locally algebra-ed topos.
Let $\mathcal{C}_{\mathbb{T}}$ be the syntactic category of an essentially algebraic theory $\mathbb{T}$, hence any category with finite limits. Let $J$ be a subcanonical coverage on $\mathcal{C}_{\mathbb{T}}$. Notice that this makes $(\mathcal{C}_{\mathbb{T}}, J)$ be a standard site and every standard site will do.
Then the sheaf topos $Sh(\mathcal{C}_{\mathbb{T}}, J)$ is the classifying topos for the geometric theory of $\mathbb{T}$-local algebras.
For $\mathcal{E}$ any topos, a local $\mathbb{T}$-algebra object in $\mathcal{E}$ is a geometric morphism
By the discussion at classifying topos this is equivalently a functor
such that
it preserves finite limits (and hence produces a $\mathbb{T}$-algebra in $\mathcal{E}$);
it sends $J$-coverings to epimorphisms; which makes it a local $\mathbb{T}$-algebra.
The pair $(\mathcal{E}, \mathcal{O}_X)$ is called a locally $\mathbb{T}$-algebra-ed topos.
All of the following notions are special cases of locally algebra-ed toposes:
The (∞,1)-category theory-version is that of