nLab pointed category

Pointed categories

Pointed categories


A category is sometimes called pointed if it has a zero object (e.g. Quillen 67, II.2 def. 4), i.e. if it has an initial object and a terminal object and they are isomorphic.

(If the morphism from the initial object to the terminal object is not necessarily an isomorphism but just a monomorphism then one speaks also of a quasi-pointed category.)

Every category of pointed objects is a pointed category in this sense, and this is probably the motivation for the terminology.

Beware that the concept of pointed objects in Cat, which may also be called “pointed categories” is more general and more restricted: these are categories with any one object singled out, and with functors between them required to preserve these objects.

Beware also that the concept of well-pointed topos is unrelated.


Canonical enrichment in pointed sets

Every locally small pointed category is naturally enriched over the category of pointed sets using the smash product as tensor product; the converse is true if and only if the enriched category has an initial object.


  • Daniel Quillen, chapter I, section 2, def. 4 of Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967, iv+156 pp.

Last revised on May 30, 2022 at 16:42:20. See the history of this page for a list of all contributions to it.