With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
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 $\mathcal{C}$ 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.
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.
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.
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.
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.
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
and a pointing of an object $v$ of $V$ is equivalent to a pointing of its underlying object in the classical sense, $1 \to U(v)$.
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.
The term appears, for instance, in:
Last revised on August 7, 2023 at 14:20:42. See the history of this page for a list of all contributions to it.