pointed object

Pointed objects


In a category CC with a terminal object, a pointed object is an object XX equipped with a global element 1X1\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 CC is the co-slice category 1/C1/C under the terminal object.

There is an obvious forgetful functor from 1/C1/C to CC. If CC has finite coproducts, this functor has a left adjoint functor which takes an object XX to the coproduct 1X1\sqcup X, equipped with its obvious point (this functor underlies the “maybe monad”). This is often written X +X_+ and called ”XX with a disjoint basepoint adjoined.” A pointed object is equivalently a module over a monad of this monad.



Zero objects and pointed categories

The category of pointed objects in any category CC 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 CC has a zero object, then every object is automatically pointed in a unique way, so that CC is equivalent to its category of pointed objects.

Closed and monoidal structure

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

[X,Y] * 1 [X,Y] [1,Y] \array{ [X,Y]_* & \rightarrow & 1\\ \downarrow && \downarrow\\ [X,Y] & \rightarrow & [1,Y]}

Here the map [X,Y][1,Y][X,Y]\to [1,Y] is induced from the point 1X1\to X, and the map 1[1,Y]1\to [1,Y] is adjunct to 111Y1\otimes 1 \to 1 \to Y. We give [X,Y] *[X,Y]_* the basepoint induced by the map 1[X,Y]1\to [X,Y] whose adjunct is 1X1Y1\otimes X \to 1 \to Y. If CC also has finite colimits, this pointed-hom has a left adjoint called the smash product, defined to be the pushout

(X1)(1Y) 1 XY XY \array{(X\otimes 1) \sqcup (1\otimes Y) & \rightarrow & 1\\ \downarrow && \downarrow\\ X\otimes Y & \rightarrow & X\wedge Y}

with the obvious basepoint. These constructions make 1/C1/C itself a closed monoidal category, which is symmetric if CC is. The unit is I +I_+, where II is the unit for the monoidal structure on CC. (The case when CC 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 CC is monoidal but not closed, the same definition of the smash product makes 1/C1/C monoidal as long as the tensor product of CC 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 CC is cartesian monoidal, but this restriction is not necessary.

Moreover, if CC is a monoidal model category with cofibrant unit, then 1/C1/C is also a monoidal model category, and the adjunction 1/CC1/C \rightleftarrows C is Quillen.

For base change functoriality of these structures see at Wirthmüller context – Examples – On pointed objects.

Kernels and cokernels

For a morphism f:ABf : A \to B into an object BB equipped with a point ptpt BBpt \stackrel{pt_B}{\to} B, its kernel ker pt B(f)ker_{pt_B}(f) is the pullback

ker pt B(f) A f pt pt B B. \array{ ker_{pt_B}(f) &\to& A \\ \downarrow && \downarrow^f \\ pt &\stackrel{pt_B}{\to}& B } \,.

The kernel is itself naturally a pointed object if AA is and if ff is a morphism of pointed objects.

Similarly, the cokernel of such a morphism is the pushout

A f B pt pt coker(f) coker(f), \array{ A &\stackrel{f}{\to}& B \\ \downarrow && \downarrow \\ pt &\stackrel{pt_{coker(f)}}{\to}& coker(f) } \,,

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:AB0 : A \to B given by Aptpt BBA \to pt \stackrel{pt_B}{\to} B.


Pointed objects are the algebras over a monad of the monad XX*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.


Revised on February 12, 2014 06:17:44 by Urs Schreiber (