In mathematics, especially category theory, Joyal's theta category is an alternative to the simplex category . It was introduced by André Joyal to give a definition of an âÂÂ-category using -sets = presheaves on instead of simplicial sets = presheaves on . Namely, in the definition of Boardman and Vogt (which is the standard definition today), an âÂÂ-category is defined as a simplicial set satisfying the weak Kan condition. In a similar way, Joyal proposed to define an âÂÂ-category as a -set satisfying the weak Kan condition.
In practice, the category is often used to define (âÂÂ, n)-categories.