nLab
pointed (infinity,1)-category

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Definition

Under a pointed (,1)(\infty,1)-category is often understood an (∞,1)-category that admits a zero object, i.e. an object which is both initial and terminal (this then is unique up to equivalence).

The terminology “pointed (,1)(\infty,1)-category”, in this sense, is commonly used, for instance, when speaking about stable (∞,1)-categories, which are such pointed (,1)(\infty,1)-categories with further properties.

If a pointed (,1)(\infty,1)-category in this sense happens to be just a 1-category, then it is a pointed category.

The same terminological caveat applies as applies to “pointed categories”:

More generally, a pointed (∞,1)-category could be taken to be a pointed object in (∞,1)Categories, i.e. an (∞,1)-category with any of its objects singled out, and with (∞,1)-functors between such pointed (,1)(\infty,1)-categories required to preserved these chosen objects.

See also

References

Last revised on February 10, 2021 at 09:11:38. See the history of this page for a list of all contributions to it.