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

Notes

References

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Further reading

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

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Category theory