Joyal's theta category

In mathematics, especially category theory, Joyal's theta category $\Theta$ is an alternative to the simplex category $\Delta$. It was introduced by André Joyal to give a definition of an ∞-category using $\Theta$-sets = presheaves on $\Theta$ instead of simplicial sets = presheaves on $\Delta$. 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 $\Theta$-set satisfying the weak Kan condition.

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

See also

• test category

References

• Theta category at the nLab
• Joyal, Andre (1997). "Disks, duality and Theta-categories" (PDF).
• Cisinski, Denis-Charles; Maltsiniotis, Georges (2011). "La catégorie $\theta$ de Joyal est une catégorie test". Journal of Pure and Applied Algebra. 215 (5): 962–982. doi:10.1016/j.jpaa.2010.07.003.

Further reading

• http://pantodon.jp/index.rb?body=Joyal_theta in Japanese