nLab pointed object in a monoidal category

Idea

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 $\mathcal{C}$ is an object $X$ equipped with a morphism $I \to X$ from the monoidal unit $I$.

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 $I \downarrow \mathcal{C}$.

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.

Examples

Example

A pointed endofunctor on a category $C$ is an endofunctor $F$ together with a natural transformation $1_C \to F$ out of the identity functor. Since the endofunctor category $[C, C]$ 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.

Example

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

Example

A pointed module is a module $M$ equipped with a morphism $R \to M$ from the ground ring $R$, where $R$ is also the unit for the tensor product of modules. Similarly, a pointed vector space is a vector space $V$ equipped with a morphism $F \to V$ from the ground field $F$, where $F$ is also the unit for the tensor product of vector spaces.

Example

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

Example

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 $V$ is a monoidal category, there is a lax monoidal change of base functor

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

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

Example

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

References

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]