Joyal's theta category

In mathematics, especially category theory, Joyal's theta category ${\displaystyle \Theta }$ is an alternative to the simplex category ${\displaystyle \Delta }$. It was introduced by André Joyal to give a definition of an ∞-category using ${\displaystyle \Theta }$-sets = presheaves on ${\displaystyle \Theta }$ instead of simplicial sets = presheaves on ${\displaystyle \Delta }$. Namely, in the definition of Boardman and Vogt (which is the standard definition today), an ∞-category is defined as a simplicial set satisfying the weak Kan condition. In a similar way, Joyal proposed to define an ∞-category as a ${\displaystyle \Theta }$-set satisfying the weak Kan condition.^[1]

In practice, the category ${\displaystyle \Theta }$ is often used to define (∞, n)-categories.

See also

• test category