nLab implicit infinity-category theory convention

Work on $\infty$-category and $(\infty,1)$-category theory often uses the implicit $\infty$-category theory convention, leaving out the “$\infty$” or “$(\infty,1)$” symbol in notation and terminology.

In this convention

• category means (∞,1)-category, and so $Cat$ means (∞,1)Cat

• topos means (∞,1)-topos, and so $Topos$ means (∞,1)Topos

• groupoid means ∞-groupoid, and so $Grpd$ means ∞Grpd

• $n$-category means (∞,n)-category

• functor means (∞,1)-functor or (∞,n)-functor for some $n \in \mathbb{N}$, depending on the codomain;

• presheaf means (∞,1)-presheaf,

• sheaf means (∞,1)-sheaf , a.k.a. ∞-stack

• limit means (∞,1)-limit

• algebraic theory means (∞,1)-algebraic theory

• algebra over an algebraic theory means ∞-algebra over an (∞,1)-algebraic theory .

Etc.

This convention is used for example by Jacob Lurie and often on nLab entries that discuss higher category theory.

(Note that this convention never applies to the names of pages in the Lab; see Naming conventions. That also means that when writing pages using this convention, one must be sure to use the appropriate “pipelinks,” such as <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos'>topos</a>.)