Pointed objects

Idea

In a category $C$ with a terminal object, a pointed object is an object $X$ equipped with a global element $1\to X$, often called its basepoint.

A pointed object is distinguished from an inhabited one in that the chosen point is structure rather than a property. In particular, a morphism of pointed objects is a morphism in the original category which preserves the points. In other words, the category of pointed objects in $C$ is the co-slice category $1/C$ under the terminal object.

There is an obvious forgetful functor from $1/C$ to $C$. If $C$ has finite coproducts, this functor has a left adjoint functor which takes an object $X$ to the coproduct $1\bigsqcup X$, equipped with its obvious point. This is often written $X_{+}$ and called ”$X$ with a disjoint basepoint adjoined.”

Examples

• Pointed topological spaces and simplicial sets are important in homotopy theory, where they are often called based.

• Pointed $n$-categories figure prominently in the delooping hypothesis; see also k-tuply monoidal n-category. In particular, a fancy name for a pointed set is a 0-tuply monoidal 0-category.

Zero objects and pointed categories

The category of pointed objects in any category $C$ with a terminal object always has a zero object, i.e. with an object which is both a terminal and initial: this is the point itself regarded as a pointed object in the unique way. A category with a zero object is sometimes called a pointed category (not to be confused with a pointed object in Cat).

Conversely, if $C$ has a zero object, then every object is automatically pointed in a unique way, so that $C$ is equivalent to its category of pointed objects.

Closed and monoidal structure

If $C$ is a closed monoidal category with finite limits and $X$ and $Y$ are pointed objects in $C$, we can consider their pointed internal-hom (the “object of basepoint-preserving maps”), defined as the pullback

Here the map $[X,Y]\to [1,Y]$ is induced from the point $1\to X$, and the map $1\to [1,Y]$ is adjunct to $1\otimes 1\to 1\to Y$. We give $[X,Y{]}_{*}$ the basepoint induced by the map $1\to [X,Y]$ whose adjunct is $1\otimes X\to 1\to Y$. If $C$ also has finite colimits, this pointed-hom has a left adjoint called the smash product, defined to be the pushout

with the obvious basepoint. These constructions make $1/C$ itself a closed monoidal category, which is symmetric if $C$ is. The unit is $I_{+}$, where $I$ is the unit for the monoidal structure on $C$. (The case when $C$ is cartesian, or at least semicartesian, is most common in the literature, but these facts are true in general. A proof can be found in Lemma 4.20 of arXiv:0710.0082.)

If $C$ is monoidal but not closed, the same definition of the smash product makes $1/C$ monoidal as long as the tensor product of $C$ preserves finite colimits in each variable separately. If not, the smash product can fail to be associative; for instance, the smash product on the ordinary category Top (without any niceness conditions imposed) is not associative.

This construction is almost always applied only when $C$ is cartesian monoidal, but this restriction is not necessary.

Moreover, if $C$ is a monoidal model category with cofibrant unit, then $1/C$ is also a monoidal model category, and the adjunction $1/C\rightleftarrows C$ is Quillen.

Kernels and cokernels

For a morphism $f:A\to B$ into an object $B$ equipped with a point $\mathrm{pt}\stackrel{{\mathrm{pt}}_B}{\to }B$, its kernel ${\ker }_{{\mathrm{pt}}_B}(f)$ is the pullback

The kernel is itself naturally a pointed object if $A$ is and if $f$ is a morphism of pointed objects.

Similarly, the cokernel of such a morphism is the pushout

which is always naturally pointed as indicated.

The notion of kernel in a category with zero morphism is obtained from this in the special case that all objects are assumed to be pointed, so that we are in a pointed category with zero-morphism $0:A\to B$ given by $A\to \mathrm{pt}\stackrel{{\mathrm{pt}}_B}{\to }B$.