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 $1\sqcup X$, equipped with its obvious point (this functor underlies the “maybe monad”). This is often written $X_+$ and called “$X$ with a disjoint basepoint adjoined.” A pointed object is equivalently a module over a monad of this monad.
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.
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.
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 Elmendorf-Mandell 07, lemma 4.20
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.
For base change functoriality of these structures see at Wirthmüller context – Examples – On pointed objects.
For a morphism $f : A \to B$ into an object $B$ equipped with a point $pt \stackrel{pt_B}{\to} B$, its kernel $ker_{pt_B}(f)$ 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 $A \to pt \stackrel{pt_B}{\to} B$.
Pointed objects are the algebras over a monad of the monad $X \mapsto X \coprod \ast$ (the “maybe monad”). (Already the unit axiom of the monad makes its algebras be pointed objects, the action axiom does not add any further condition in this case.)
Notice that if sufficient colimits exist in the first place, then this functor is trivially an accessible functor, hence an accessible monad. This makes categories of pointed objects inherit good properties from the ambient category, see at accessible monad – Categories of algebras.
The classifying topos for pointed object is the presheaf topos $PSh((FinSet_\ast)^{op})$ on the opposite category of pointed finite sets. See at classifying topos for the theory of objects for more on this.