nLab pointed object in a monoidal category


Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



Just as the classical notion of pointed objects refers to morphisms whose domain is terminal, which in the context or doctrine of cartesian monoidal categories is the monoidal unit, so one can generalize such “pointing”: a pointed object in a monoidal category 𝒞\mathcal{C} is an object XX equipped with a morphism IXI \to X from the monoidal unit II.

A morphism between a pair of monoidally-pointed objects is then typically taken to be a morphism of the underlying objects which respects these “points” under precomposition. This means that the category of monoidally-pointed objects is the coslice category I𝒞I \downarrow \mathcal{C}.

Therefore, yet more generally, one might regard any coslice category as a category of generalized-pointed objects. But the coslice under a monoidal unit has further good properties, such as itself canonically inheriting the structure of a monoidal category.



A pointed endofunctor on a category CC is an endofunctor FF together with a natural transformation 1 CF1_C \to F out of the identity functor. Since the endofunctor category [C,C][C, C] may be viewed as a strict monoidal category whose monoidal unit is the identity functor, this is an example of a pointed object in a monoidal category.


A pointed abelian group is an abelian group AA equipped with a morphism A\mathbb{Z} \to A, where \mathbb{Z} is the unit for the tensor product of abelian groups.


A pointed module is a module MM equipped with a morphism RMR \to M from the ground ring RR, where RR is also the unit for the tensor product of modules. Similarly, a pointed vector space is a vector space VV equipped with a morphism FVF \to V from the ground field FF, where FF is also the unit for the tensor product of vector spaces.


A bi-pointed set is a pointed set SS equipped with a morphism 𝔹S\mathbb{B} \to S from the boolean domain 𝔹\mathbb{B}, where the smash product is the tensor product and the boolean domain is the tensor unit of the monoidal category of pointed sets.


Under change of base of enrichment, pointed objects in the current sense may be compared to pointed objects in the classical sense. For example, if VV is a monoidal category, there is a lax monoidal change of base functor

U=V(I,):VSetU = V(I, -): V \to Set

and a pointing of an object vv of VV is equivalent to a pointing of its underlying object in the classical sense, 1U(v)1 \to U(v).


If CC is a monoidal category, then C opC^{op} acquires a monoidal category structure as well, and the coslice IC opI \downarrow C^{op} of monoidally pointed objects is equivalent to the slice CIC \downarrow I. This can be important in practice. See for example the discussion at affine space which relates the definition of affine space to the slice Vect k/kVect_k/k, and see even more particularly the discussion of the closed monoidal structure of affine spaces.


The term appears, for instance, in:

  • Paul-André Melliès, Nicolas Tabareau & Christine Tasson, p. 4 of: An Explicit Formula for the Free Exponential Modality of Linear Logic, in: Automata, Languages and Programming. ICALP 2009, Lecture Notes in Computer Science, 5556, Springer (2009) [doi:10.1007/978-3-642-02930-1_21]

Last revised on August 7, 2023 at 14:20:42. See the history of this page for a list of all contributions to it.