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.
Let $\mathcal{C}$ be a category and let $X \in \mathcal{C}$ be an object.
The slice category $\mathcal{C}_{/X}$ is the category whose
objects are morphisms $\array{A \\ \downarrow \\ X}$ in $\mathcal{C}$;
morphisms are commuting triangles $\array{ A && \longrightarrow && B \\ & {}_{}\searrow && \swarrow \\ && X}$ in $\mathcal{C}$.
Dually, the coslice category $\mathcal{C}^{X/}$ is the category whose
objects are morphisms $\array{X \\ \downarrow \\ A}$ in $\mathcal{C}$;
morphisms are commuting triangles $\array{ && X \\ & \swarrow && \searrow \\ A && \longrightarrow && B }$ in $\mathcal{C}$.
There is the canonical forgetful functor
given by forgetting the morphisms to/from $X$.
We here focus on this class of examples:
For $\mathcal{C}$ a category with terminal object $\ast$, the coslice category (def. 1) $\mathcal{C}^{\ast/}$ is the corresponding category of pointed objects: its
objects are morphisms in $\mathcal{C}$ of the form $\ast \overset{x}{\to} X$ (hence an object $X$ equipped with a choice of point; i.e. a pointed object);
morphisms are commuting triangles of the form
(hence morphisms in $\mathcal{C}$ which preserve the chosen points).
The basic example are pointed sets.
pointed topological spaces and pointed simplicial sets are important in homotopy theory (where they are often called based) for instance for the discussion of homotopy fibers, loop space objects etc. See also at classical model structure on pointed topological spaces, which makes them be models for pointed homotopy types.
Pointed $n$-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.
Let $\mathcal{C}$ be a category with terminal object and finite colimits. Then the forgetful functor $\mathcal{C}^{\ast/} \to \mathcal{C}$ from its category of pointed objects, def. 2, has a left adjoint given by forming the disjoint union (coproduct) with a base point (“adjoining a base point”), this is denoted by
In a category of pointed objects $\mathcal{C}^{\ast/}$, def. 2, the terminal object coincides with the initial object, both are given by $\ast \in \mathcal{C}$ itself, pointed in the unique way.
In this situation one says that $\ast$ is a zero object and that $\mathcal{C}^{\ast/}$ is a pointed category.
It follows that also all hom-sets $\mathcal{C}^{\ast/}(X,Y)$ of $\mathcal{C}^{\ast/}$ are canonically pointed sets, pointed by the zero morphism
Conversely, if $\mathcal{C}$ has a zero object, then every object is automatically pointed in a unique way, so that $\mathcal{C}$ is equivalent to its category of pointed objects.
Let $\mathcal{C}$ be a category with all limits and colimits. Then also the category of pointed objects $\mathcal{C}^{\ast/}$, def. 2, has all limits and colimits.
Moreover:
the limits are the limits of the underlying diagrams in $\mathcal{C}$, with the base point of the limit induced by its universal property in $\mathcal{C}$;
the colimits are the limits in $\mathcal{C}$ of the diagrams with the basepoint adjoined.
It is immediate to check the relevant universal property. For details see at slice category – limits and colimits.
Given two pointed objects $(X,x)$ and $(Y,y)$, then:
their product in $\mathcal{C}^{\ast/}$ is simply $(X\times Y, (x,y))$;
their coproduct in $\mathcal{C}^{\ast/}$ has to be computed using the second clause in prop. 1: since the point $\ast$ has to be adjoined to the diagram, it is given not by the coproduct in $\mathcal{C}$, but by the pushout in $\mathcal{C}$ of the form:
This is called the wedge sum operation on pointed objects.
Generally for a set $\{X_i\}_{i \in I}$ in $Top^{\ast/}$
For $X$ a CW-complex, then for every $n \in \mathbb{N}$ the quotient of its $n$-skeleton by its $(n-1)$-skeleton is the wedge sum, def. 1, of $n$-spheres, one for each $n$-cell of $X$:
For $\mathcal{C}^{\ast/}$ a category of pointed objects with finite limits and finite colimits, the smash product is the functor
given by
hence by the pushout in $\mathcal{C}$
In terms of the wedge sum from def. 1, this may be written concisely as
These two operations are ubiquituous in stable homotopy theory:
symbol | name | category theory |
---|---|---|
$X \vee Y$ | wedge sum | coproduct in $\mathcal{C}^{\ast/}$ |
$X \wedge Y$ | smash product | tensor product in $\mathcal{C}^{\ast/}$ |
For $X, Y \in Top$, with $X_+,Y_+ \in Top^{\ast/}$, def. 3, then
$X_+ \vee Y_+ \simeq (X \sqcup Y)_+$;
$X_+ \wedge Y_+ \simeq (X \times Y)_+$.
By example 1, $X_+ \vee Y_+$ is given by the colimit in $Top$ over the diagram
This is clearly $A \sqcup \ast \sqcup B$. Then, by definition 4
Let $\mathcal{C}^{\ast/} = Top^{\ast/}$ be pointed topological spaces. Then
denotes the standard interval object $I = [0,1]$, with a djoint basepoint adjoined, def. 3. Now for $X$ any pointed topological space, then
is the reduced cylinder over $X$: the result of forming the ordinary cyclinder over $X$, and then identifying the interval over the basepoint of $X$ with the point.
(Generally, any construction in $\mathcal{C}$ properly adapted to pointed objects $\mathcal{C}^{\ast/}$ is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)
Just like the ordinary cylinder $X\times I$ receives a canonical injection from the coproduct $X \sqcup X$ formed in $Top$, so the reduced cyclinder receives a canonical injection from the coproduct $X \sqcup X$ formed in $Top^{\ast/}$, which is the wedge sum from example 1:
Given a morphism $f \colon X \longrightarrow Y$ in a category of pointed objects $\mathcal{C}^{\ast/}$, def. 2, with finite limits and colimits,
its fiber or kernel is the pullback of the point inclusion
its cofiber or cokernel is the pushout of the point projection
In the situation of def. 5, both the pullback as well as the pushout are equivalently computed in $\mathcal{C}$. For the pullback this is the first clause of prop. 1. The second clause says that for computing the pushout in $\mathcal{C}$, first the point is to be adjoined to the diagram, and then the colimit over the larger diagram
be computed. But one readily checks that in this special case this does not affect the result. (The technical jargon is that the inclusion of the smaller diagram into the larger one in this case happens to be a final functor.)
Let $\mathcal{C}$ be a closed monoidal category with finite limits.
For $X, Y \in \mathcal{C}^{\ast}$ two pointed objects in $\mathcal{C}$, their pointed mapping space
(the “object of basepoint-preserving maps”), is the pullback
where the morphism $[X,Y]\to [1,Y]$ is induced from the point $\ast\to X$, and the morphism $\ast\to [\ast,Y]$ is the adjunct to $\ast \otimes \ast \to \ast \to Y$.
Regard $[X,Y]_*$ as a pointed object with basepoint induced by the map $\ast\to [X,Y]$ whose adjunct is $\ast \otimes X \to \ast \to Y$.
Let $\mathcal{C}$ be a closed monoidal category with finite limits and with finite colimits.
For every pointed object $X \in \mathcal{C}^{\ast}$ the operation of forming the pointed mapping space out of $X$, def. 6, and the operation of forming the smash product with $X$, def. 4 form a pair of adjoint functors
This makes $\mathcal{C}^{\ast/}$ itself a closed monoidal category, which is symmetric if $\mathcal{C}$ is. The tensor unit is $I_+$ (def. 3) for $I$ the unit for the monoidal structure on $\mathcal{C}$.
(Elmendorf-Mandell 07, lemma 4.20)
The case when $\mathcal{C}$ is cartesian, or at least semicartesian, is most common in the literature.
If $\mathcal{C}$ is monoidal but not closed, the same definition 4 of the smash product makes $\mathcal{C}^{\ast/}$ monoidal as long as the tensor product of $\mathcal{C}$ 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.
For base change functoriality of these structures see at Wirthmüller context – Examples – On pointed objects.
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.
Let $\mathcal{C}$ be a model category and let $X \in \mathcal{C}$ be an object. Then both the slice category $\mathcal{C}_{/X}$ as well as the coslice category $\mathcal{C}^{X/}$, def. 1, carry model structures themselves – the model structure on a (co-)slice category, where a morphism is a weak equivalence, fibration or cofibration iff its image under the forgetful functor $U$ is so in $\mathcal{C}$.
In particular the category $\mathcal{C}^{\ast/}$ of pointed objects, def. 2, in a model category $\mathcal{C}$ becomes itself a model category this way.
The model strcuture as claimed is immediate by inspection.
For $\mathcal{C} = Top_{Quillen}$, the classical model structure on topological spaces, then the model structure on pointed topological spaces induced via prop. 3 we call the classical model structure on pointed topological spaces $Top_{Quillen}^{\ast/}$. Its homotopy category of a model category is the classical pointed homotopy theory $Ho(Top^{\ast/})$.
The fibrant objects in the pointed model structure $\mathcal{C}^{\ast/}$, prop. 3, are those that are fibrant as objects of $\mathcal{C}$.
But the cofibrant objects in $\mathcal{C}^{\ast}$ are now those for which the basepoint inclusion is a cofibration in $X$.
For $\mathcal{C}^{\ast/} = Top^{\ast/}$, then the corresponding cofibrant pointed topological spaces are tyically referred to as spaces with non-degenerate basepoints. Notice that the point itself is cofibrant in $Top_{Quillen}$, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.
For $\mathcal{C}$ any model category, with $\mathcal{C}^{\ast/}$ its pointed model structure according to prop. 3, then the corresponding homotopy category is, by remark 1, canonically enriched in pointed sets, in that its hom-functor is of the form
If $\mathcal{C}$ is a monoidal model category with cofibrant tensor unit, then the pointed model structure on $\mathcal{C}^{\ast/}$ (prop. 3) is also a monoidal model category, and the smash product$\dashv$mapping space adjunction of prop. 2 is a Quillen adjunction
Francis Borceux, Dominique Bourn, Mal’cev, Protomodular, Homological and Semi-Abelian Categories , Kluwer Dordrecht 2004.
Anthony Elmendorf, Michael Mandell, Permutative categories, multicategories, and algebraic K-theory, Algebraic & Geometric Topology 9 (2009) 2391-2441. (arXiv:0710.0082)