pointed derivator

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#### Homotopy theory
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[[!include homotopy - contents]]
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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# Pointed derivators
* table of contents
{: toc}

## Idea

A [[derivator]] is *pointed* if it has a [[zero object]] in the sense appropriate to a derivator.

## Definition

A derivator $D\colon Dia^{op} \to Cat$ is **pointed**, or has a **zero object**, if each [[category]] $D(X)$ has a zero object.

Note that these zero objects are automatically preserved by all the [[functors]] $u_!,u^*,u_*$, since they are all [[adjoint functor|adjoints]] on one side or the other.

## Properties

This simple definition hides a lot of structure.

### Relative diagram categories

If $D$ is a pointed derivator and $i\colon A\hookrightarrow B$ is a [[full subcategory]], we write $D(B,A)$ for the full subcategory of $D B$ on those diagrams $X\in D B$ such that $i^*X$ is null.

+-- {: .num_lemma #SieveRel}
###### Lemma
If $i$ is the inclusion of a [[sieve]], then $i_*\colon D(A) \to D(B)$ is [[full and faithful functor|fully faithful]] and its [[essential image]] is $D(B,B\setminus A)$.  Similarly, if $i$ is a cosieve, then $i_!$ identifies $D(A)$ with $D(B,B\setminus A)$.
=--
+-- {: .proof}
###### Proof
Suppose $i$ is a sieve.  Since $i$ is in particular fully faithful, so is $i_*$ by general arguments (see [[homotopy exact square]]).  Moreover, since $i$ is a sieve, the square
$$\array{ \emptyset & \to & A\\
\downarrow & & \downarrow^i\\
B\setminus A & \underset{j}{\to} & B}
$$
is homotopy exact (where $j$ is the inclusion).  Therefore $j^* i_* X$ is terminal for any $X\in D(A)$, so that $i_*$ lands inside $D(B,B\setminus A)$.  Conversely, if $X\in D(B,B\setminus A)$, then the map $X \to i_* i^* X$ is an isomorphism on $A$ (since $i_*$ is fully faithful) and on $B\setminus A$ (since both are terminal there), hence is itself an isomorphism; so $X$ is in the essential image of $i_*$.
=--

The following theorem is stated as Proposition 7.4 of [Heller's paper](#HellerStable) "Stable homotopy theories and stabilization".

+-- {: .num_theorem #RelRefl}
###### Theorem
$D(B,A)$ is a [[reflective subcategory|reflective]] and [[coreflective subcategory|coreflective]] subcategory of $D(B)$.
=--
+-- {: .proof}
Let $M i$ denote the "codirected [[mapping cylinder]]" of $i\colon A \hookrightarrow B$, by which we mean the full subcategory of $B\times I$ on the objects $(x,0)$ for $x\in B$ and $(i(x),1)$ for $x\in A$ (here $I$ is the [[interval category]] $(0\to 1)$).  This is also the same as the [[cocomma object]] $(B \uparrow i)$.  Write $B \overset{u}{\to} M i \overset{p}{\to} B$ for the evident inclusion and projection.

Consider the adjunction
$$ D(M i) \underoverset{p^*}{p_!}{\rightleftarrows} D(B). $$
We claim that this restricts to an adjunction
\[ D(M i,A) \underoverset{p^*}{p_!}{\rightleftarrows} D(B,A) \label{ppadj} \]
where $A$ denotes the subcategory of $M i$ on the objects $(i(x),1)$.  Certainly $p^*$ takes $D(B,A)$ into $D(M i,A)$, since $i\colon A \hookrightarrow B$ is the composite $A \hookrightarrow M i \overset{p}{\to} B$.  On the other hand, the square
$$\array{A & \hookrightarrow & M i \\
\downarrow & & \downarrow^p\\
A & \underset{i}{\to} & B }
$$
is [[homotopy exact square|homotopy exact]], so $p_!$ takes $D(M i,A)$ into $D(B,A)$.

Now since $u$ is the inclusion of a [[sieve]], by the previous lemma, $u_*$ identifies $D(B)$ with $D(M i,A)$.  Therefore, the inclusion $D(B,A)\hookrightarrow D(B)$ can be identified with the restricted right adjoint $p^*\colon D(B,A) \to D(M i,A)$, which therefore has a left adjoint $p_!$.  Explicitly, the left adjoint is $p_! u_*$.

The proof of coreflectivity is of course dual, using the directed mapping cylinder instead of the codirected one.
=--

Note that if $i'\colon A'\hookrightarrow B'$ is another full inclusion and
$$\array{A & \overset{f}{\to} & A'\\
  ^i\downarrow && \downarrow^{i'}\\
  B& \underset{g}{\to} & B'}$$
commutes, then we have a restriction $(g,f)^*\colon D(B',A') \to D(B,A)$, which has a left adjoint $(g,f)_!$ given by the composite
$$ D(B,A) \hookrightarrow D(B) \overset{g_!}{\to} D(B') \overset{L_{B',A'}}{\to} D(B',A') $$
where $L_{B,A}\colon D(B) \to D(B,A)$ denotes the reflection.  It also has a right adjoint defined dually.

### Pointed exactness

Define a **category with zeros** to be a category $B$ equipped with a full subcategory $B_0$.  By the preceding remarks, any pointed derivator gives rise to a contravariant pseudofunctor defined on the 2-category of categories with zeros (and functors preserving the specified subcategories), all of whose transition functors have both adjoints.  It is natural to look for exactness conditions, analogous to (Der4) and the characterization of [[homotopy exact squares]], which apply to these adjoints.

Since the Beck-Chevalley transformation relating composites of these adjoints is the composite of the corresponding transformation for the unpointed diagram and a Beck-Chevalley transformation relating the restriction functors to the reflections $D(B) \to D(B,B_0)$, we clearly need to study when restriction commutes with these reflections.  This is the purpose of the following definition and lemma.

+-- {: .num_defn #LocallyNullFinal}
###### Definition
A functor $f\colon A \to B$ between categories with zeros is **locally null-final** if for every $a\in A$, every $b_0 \in B_0$, and every morphism $\phi\colon b_0 \to f(a)$, the category of triples $(a_0\in A_0, a_0 \xrightarrow{\alpha} a, b_0 \xrightarrow{\beta} f(a_0))$ such that $\phi = f(\alpha).\beta$ has a contractible nerve.
=--

+-- {: .num_lemma #LocallyNullFinalLemma}
###### Lemma
If $f\colon A \to B$ is [locally null-final](#LocallyNullFinal) and $D$ is a pointed derivator, then the functors $f^*\colon D(B) \to D(A)$ and $f^*\colon D(B,B_0) \to D(A,A_0)$ commute with the reflections of $D(B)$ and $D(A)$ into $D(B,B_0)$ and $D(A,A_0)$, respectively.  In other words, the canonical [[natural transformation]]

$$
  \array{
    D(A) & \xleftarrow{f^*} & D(B)
    \\
    ^{L_A}\downarrow & \seArrow & \downarrow^{L_B}
    \\
    D(A,A_0) & \xleftarrow{f^*} & D(B,B_0)
  }
$$

is an [[isomorphism]].

=--

+-- {: .proof}
###### Proof
Recall that $L_A$ is computed as the composite $D(A) \xrightarrow{u_*} D(M_A, A_0) \xrightarrow{p_!} D(A,A_0)$, where $M_A$ is the codirected mapping cylinder of $A_0\to A$.  We can therefore factor the above square as
$$\array{
  D(A) & \xleftarrow{f^*} & D(B) \\
  ^{u_*}\downarrow & & \downarrow^{u_*} \\
  D(M_A,A_0) & \xleftarrow{f^*} & D(M_B,B_0)\\
  ^{p_!}\downarrow & & \downarrow^{p_!} \\
  D(A,A_0) & \xleftarrow{f^*} & D(B,B_0)}
$$
It suffices, therefore, to show that the squares
$$\array{A & \xrightarrow{u} & M_A \\
  ^f \downarrow & & \downarrow^f \\
  B & \xrightarrow{u} & M_B}
\qquad\text{and}\qquad
\array{M_A & \xrightarrow{f} & M_B\\
  ^p\downarrow & & \downarrow^p\\
  A & \xrightarrow{f} & B}
$$
are homotopy exact.

For the first square, consider first a $b\in B$ and $a\in A\subset M_A$ and a $\varphi\colon f(a)\to b$ in $B\subset M_B$.  The category of triples $(a', a\to a', f(a')\to b)$ which compose to $\varphi$ is contractible, since $(a, id_a, \varphi)$ is an initial object.  Second, we should consider a $b\in B$ and an $a_0\in A_0 \subset M_A$, and a $\varphi\colon f(a_0)\to b$ in $B_0 \subset M_B$ --- but by definition of $M_B$, no such morphism $\varphi$ can exist.  Thus, the first square is always homotopy exact.

In fact, however, the preceeding argument is unnecessary, because we have seen that with their targets restricted as above, the functors $u_*$ are equivalences, and the [[mate]] of an isomorphism with respect to any pair of adjoint equivalences is again an isomorphism.  Note that the fact that $u_*$ is an equivalence was already used in making the assertion that $p_! u_*$ is a left adjoint to $D(A,A_0) \hookrightarrow D(A)$, and therefore the inverse of the top square already factors into the definition of the transformation appearing in the statement of the lemma.  (This may reassure a reader who was worried about the fact that the canonical transformation in the first square goes the "wrong direction".)

For the second square, consider first an $a\in A$ and a $b\in B\subset M_B$, and a $\varphi\colon b \to f(a)$.  The category we must investigate containts two types of objects:

* triples $(a'\in A, b\to f(a'), a' \to a)$ which compose to $\varphi$, and
* triples $(a_0 \in A_0, b\to f(a_0), a_0 \to a)$ which compose to $\varphi$.

By definition of $M_A$, the morphisms between these objects are the obvious ones, except that there are no morphisms from the second type to the first.  Now the full subcategory on the first type of object is coreflective, since $A_0$ is coreflective in $M_A$.  And that full subcategory is contractible, since $(a, \varphi, id_a)$ is a terminal object.  Thus, since adjunctions induce homotopy equivalences of nerves, the category in question is also contractible.

Finally, consider the case of an $a\in A$ and a $b_0\in B_0 \subset M_B$, and a $\varphi\colon b_0 \to f(a)$.  The category in question consists of triples $(a_0\in A_0, b_0 \to f(a_0), a_0\to a)$ which compose to $\varphi$ (since there are no morphisms in $M_B$ from $b_0\in B_0$ to anything in the image of $A\subset M_A$).  But this is precisely the category asserted to be contractible in the assumption that $f$ is locally null-final.
=--

+-- {: .num_theorem #HomotopyExactWithZeros1}
###### Theorem
If a square
$$\array{ I & \xrightarrow{f} & J \\
  ^h\downarrow & \swArrow & \downarrow^k\\
  K & \xrightarrow{g} & L}$$
of categories with zeros has the properties that

1. it is [[homotopy exact square|homotopy exact]] as a square of categories, when the full subcategories are ignored, and

1. $g$ is [locally null-final](#LocallyNullFinal),

then for any pointed derivator, the induced transformation
$$\array{ D(I,I_0) & \xleftarrow{f^*} & D(J,J_0) \\
  ^{h_!}\downarrow & \seArrow & \downarrow^{k_!}\\
  D(K,K_0) & \xleftarrow{g^*} & D(L,L_0)}$$
is an isomorphism.
=--
+-- {: .proof}
###### Proof
By definition, the displayed functors $h_!$ and $k_!$ are obtained by applying the $h_!$ and $k_!$ of a derivator followed by the reflection into the relative diagram categories.  Thus the given square factors into
$$\array{ D(I,I_0) & \xleftarrow{f^*} & D(J,J_0) \\
  ^{h_!}\downarrow & \seArrow & \downarrow^{k_!}\\
  D(K)  & \xleftarrow{g^*} & D(L)\\
  ^{L_K}\downarrow & \seArrow & \downarrow^{L_L}\\
  D(K,K_0) & \xleftarrow{g^*} & D(L,L_0)}$$
in which the first square is an isomorphism by the first assumption, and the second square by the second assumption and
Lemma \ref{LocallyNullFinalLemma}.

=--

For example, we can conclude:

+-- {: .num_cor #LocallNullFinalFF}
###### Corollary
If $f\colon A\to B$ is a fully faithful and locally null-final functor between categories with zeros, then for any pointed derivator $D$, the functor $f_!\colon D(A,A_0) \to D(B,B_0)$ is fully faithful.
=--
+-- {: .proof}
###### Proof
The square
$$\array{ A & \xrightarrow{id} & A \\
  ^{id}\downarrow & \swArrow & \downarrow^f\\
  A & \xrightarrow{f} & B}$$
satisfies the hypotheses of the previous theorem.
=--

Theorem \ref{HomotopyExactWithZeros1} is not best possible, however.  Its two conditions characterize when the two squares into which the Beck-Chevalley transformation factors are separately isomorphisms.  However, it might happen that the composite is an isomorphism even though one or the other of the transformations is not separately an isomorphism.

In particular, the condition of "local null-finality" on the bottom morphism $g$ uses no information about the categories $I$ and $J$.  If we know some things about them, then we can correspondingly weaken this condition.

+-- {: .num_theorem #HomotopyExactWithZeros2}
###### Theorem
Suppose given a square
$$\array{ I & \xrightarrow{f} & J \\
  ^h\downarrow & \swArrow & \downarrow^k\\
  K & \xrightarrow{g} & L}$$
of categories with zeros, where $L$ is equipped with a full subcategory $\hat{L}_0 \subseteq L_0$ of its category of zeros.  Write $\check{L}_0 = L_0 \setminus \hat{L}_0$, and suppose the following.

1. The square is [[homotopy exact square|homotopy exact]] as a square of categories, when the categories of zeros are ignored.

1. For any pointed derivator $D$, the functor $k_!\colon D(J) \to D(L)$ maps $D(J,J_0)$ into $D(L,\hat{L}_0)$.  For instance, this is the case if for each $z\in \hat{L}_0$, the category $k/z$ has a terminal object lying in $J_0$.

1. For each $x\in L\setminus L_0$ and $y\in \hat{L}_0$ and morphism $x\xrightarrow{\varphi} y$ in $L$, the following category has a contractible nerve: its objects are triples $(z\in L_0, x\xrightarrow{\alpha} z, z\xrightarrow{\beta} y)$ such that $\beta\alpha=\varphi$, and its morphisms are morphisms $z_1\to z_2$ in $L$ commuting with the given morphisms *and* such that either $z_1 \in \hat{L}_0$ or $z_2 \notin \hat{L}_0$.

1. For every $x\in \check{L}_0$ and $y\in K$ and $x \xrightarrow{\varphi} g(y)$ in $L$, the category of triples $(z\in K_0 \cap g^{-1}(\check{L}_0), x\xrightarrow{\alpha} g(z), z \xrightarrow{\beta} y)$ such that $g(\beta).\alpha= \varphi$ has a contractible nerve ("relative local null-finality").

Then for any pointed derivator, the induced transformation
$$\array{ D(I,I_0) & \xleftarrow{f^*} & D(J,J_0) \\
  ^{h_!}\downarrow & \seArrow & \downarrow^{k_!}\\
  D(K,K_0) & \xleftarrow{g^*} & D(L,L_0)}$$
is an isomorphism.
=--

+-- {: .proof}
###### Proof
**(Sketch)** Because of the second condition in the theorem, instead of the reflection $D(L) \to D(L,L_0)$ it will suffice to consider the reflection $D(L,\hat{L_0}) \to D(L,L_0)$.  We use the third condition to give an alternate way to compute this reflection, and the fourth condition to ensure that restriction along $g$ commutes with this reflection.

Let $\hat{M}_L$ be the codirected mapping cylinder of $\check{L}_0 \hookrightarrow L$.  If $L \xrightarrow{u} \hat{M}_L \xrightarrow{p} L $ are as before, then $u_*$ identifies $D(L,\hat{L}_0)$ with the subcategory of $D(\hat{M}_L)$ consisting of diagrams which are zero on the copy of $\check{L}_l$ that is the target end of the mapping cylinder and on the copy of $\hat{L}_0$ inside the copy of $L$ that is the source end; call this subcategory $D(\hat{M}_L, \hat{L}_0 \cup \check{L}_0)$.

Now if the adjunction $p_! \colon D(\hat{M}_L) \rightleftarrows D(L) \;: p^*$ restricts to an adjunction $D(\hat{M}_L, \hat{L}_0 \cup \check{L}_0)\rightleftarrows D(L,L_0)$, we will be able to compute the reflection $D(L,\hat{L}_0) \to D(L,L_0)$ as $p_! u_*$, as before.  This will follow if the following square is exact:
$$\array{ \hat{L}_0 \cup \check{L}_0 & \to & \hat{M}_L \\
  \downarrow & & \downarrow \\
  L_0 & \to & L } $$
where $\hat{L}_0 \cup \check{L}_0$ denotes the same full subcategory of $\hat{M}_L$ as above.  Exactness of this square can be verified to be equivalent to the third condition in the theorem.

Finally, modifying the proof of Lemma \ref{LocallyNullFinalLemma}, we see that the fourth condition in the theorem implies commutativity of $g^*$ with this reflection $p_! u_*$.  Therefore, the theorem follows as before.
=--

If $\hat{L}_0 = \emptyset$, then the second and third conditions of Theorem \ref{HomotopyExactWithZeros2} are vacuous and it reduces to Theorem \ref{HomotopyExactWithZeros1}.  At the other extreme, if $\hat{L}_0 = L_0$, then the second and fourth conditions of Theorem \ref{HomotopyExactWithZeros2} are vacuous and the third becomes "$k_!$ maps $D(J,J_0)$ into $D(L,L_0)$."

### Extraordinary inverse images

+-- {: .num_theorem #EOInv}
###### Theorem
A derivator $D$ is pointed if and only if whenever $u\colon A\to B$ is a [[sieve]] in $Dia$, the functor $u_* \colon D(A) \to D(B)$ has a right adjoint $u^!$, and dually whenever $u$ is a cosieve, the functor $u_!$ has a left adjoint $u^?$.
=--
+-- {: .proof}
The "if" direction of this is easy, since $u\colon \emptyset \to B$ is always a sieve, while $u_*\colon * \to U(B)$ assigns the terminal object of $D(B)$.  Thus, if $u_*$ has a further right adjoint, it must preserve colimits, and in particular its value on the unique object of $D(\emptyset)$ must be an initial object (as well as a terminal one).

Conversely, if $D$ is pointed and $u$ is a sieve, then by Lemma \ref{SieveRel}, $u_*$ identifies $D(A)$ with $D(B,B\setminus A)$.  But by Theorem \ref{RelRefl}, the inclusion $D(B,B\setminus A)\hookrightarrow D(B)$ has a right adjoint.  The other case is dual.
=--

The functors $u^!$ and $u^?$ are sometimes referred to as *extraordinary* and *co-extraordinary inverse image* functors.  Our proof shows that $u^? = u^* p_! v_*$, where $v$ and $p$ are respectively the inclusion of $B$ in the codirected mapping cylinder of $B\setminus A \hookrightarrow B$, and its projection to $B$.

In the literature, the existence of these functors is often taken as the definition of when a derivator is pointed.  The equivalence of the two definitions is a "super-difficult" exercise in [Maltsionitis' notes](http://people.math.jussieu.fr/~maltsin/Seville/Lecture_III_Derivators.pdf).

### Loop and suspension

In a pointed derivator, we have a [[suspension]] functor $\Sigma\colon D 1 \to D 1$ defined as the composite
$$ D 1 \overset{a_*}{\to} D \Gamma \overset{a b c_!}{\to} D \square \overset{d^*}{\to} D 1$$
where $\square$ denotes the category
$$ \array{ a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d } $$
and $\Gamma$ its full subcategory on $\{a,b,c\}$.

Equivalently, the suspension can be defined as the composite
$$ D 1 = D(1,\emptyset) \overset{(a,\emptyset)_!}{\to} D(\square,b c) \overset{(d,\emptyset)^*}{\to} D(1,\emptyset) = D 1.$$
This follows because we have $D 1 \simeq D(\Gamma,b c)$, and thus $(a,\emptyset)_!$ is isomorphic to $(a b c,b c)_!\colon D(\Gamma,b c) \to D(\square,b c)$, which by definition is $L_{\square,b c} \circ a b c_!$.  But $a b c_!$ is fully faithful since $a b c\colon \Gamma \hookrightarrow \square$ is, and so it already takes $D(\Gamma,b c)$ into $D(\square,b c)$.  Thus $L_{\square,b c}$ is doing nothing, so $(a,\emptyset)_!$ is essentially just $a b c_!$ which appeared in our previous definition of $\Sigma$.  (The other functor $a_*$ essentially implements the equivalence $D 1 \simeq D(\Gamma,b c)$.)

Similarly, we have a [[loop space object]] functor $\Omega\colon D 1 \to D 1$ defined as the composite
$$ D 1 \overset{d_!}{\to} D \Gamma^{op} \overset{b c d_*}{\to} D \square \overset{a^*}{\to} D 1.$$
where $\Gamma^{op}$ is the full subcategory of $\square$ on $\{b,c,d\}$, and it can equivalently be given as the composite
$$ D 1 = D(1,\emptyset) \overset{(d,\emptyset)_*}{\to} D(\square,bc) \overset{(a,\emptyset)^*}{\to} D(1,\emptyset) = D 1.$$
The description in terms of relative diagram categories makes it clear that $\Sigma \dashv \Omega$.

We can also describe $\Sigma$ and $\Omega$ in terms of the extraordinary inverse image functors.  Since the square
$$ \array{ \Gamma & \to & \Gamma \\
^r\downarrow & & \downarrow^{a b c}\\
* & \underset{d}{\to} & \square}$$
is homotopy exact, the composite $d^* a b c_!$ in our first definition is the same as $r_!$.  The other functor $a_*$ can in turn be decomposed as a composite $a b_* s_*$, where $s\colon * \to I$ is the inclusion of $s$ into the interval category $(s\to t)$, and $ab\colon I\to \Gamma$ is the inclusion into $\{a,b\}$.

Therefore we have $\Sigma \cong r_! a b_* s_*$.  But $\Gamma$ can be identified with $M s$ and $a b$ with the inclusion $u\colon I \hookrightarrow M s$, while $r$ factors as $M s \overset{p}{\to} I \overset{q}{\to} *$ so that $r_! \cong q_! p_!$.  But finally, the square
$$\array{* & \overset{t}{\to} & I\\
\downarrow & & \downarrow^q\\
*& \to & *}$$
is homotopy exact, so $q_! \cong t^*$ and thus
$$\Sigma \cong t^* p_! u_* s_* \cong t^? s_*.$$
Dually, we can identify $\Omega \cong s^! t_!$, from which it again follows directly that $\Sigma \dashv \Omega$.

### Enrichment over pointed sets

Every pointed derivator can be canonically "[[enriched derivator|enriched]]" over [[pointed sets]], in the sense that we can extend it to a functor
$$D': \; Set_* Cat \; \to\; Set_* CAT$$
from small $Set_*$-enriched categories to large ones, satisfying analogues of the derivator axioms.  Specifically, if $C$ is a $Set_*$-category, define $\bar{C}$ to be $C$ with a [[zero object]] adjoined.  (Note that zero objects are [[absolute colimits]] in $Set_*$-categories.)  Then set $D'(C) = D(\bar{C},0)$, the "relative diagram category" as defined above, i.e. the full subcategory of $D(\bar{C})$ on those diagrams which send the zero object to zero.

Any $Set_*$-functor automatically preserves zero objects (since an object $x$ of a $Set_*$-category is zero iff its identity $id_x$ is the basepoint of the pointed homset $hom(x,x)$).  In particular, any $Set_*$-functor $u:A\to B$ induces a relative functor $(u,0):(\bar{A},0) \to (\bar{B},0)$, and hence a pullback $u^* = (u,0)^*:D'(B) \to D'(A)$.  According to the general theory of relative diagram categories, this functor has left and right adjoints $u_!$ and $u_*$, obtained from left and right extension along $u$ followed by reflection or coreflection from $D(\bar{B})$ into $D(\bar{B},0)$.  In fact, however, in this case the reflection/coreflection is unnecessary: since the comma category $(u,0)/0$ has a terminal object $0$, the functor $u_!$ already maps $D(\bar{A},0)$ into $D(\bar{B},0)$.

Thus we have a $Set_*$-analogue of the derivator axiom (Der3), existence of adjoints.  For the enriched axiom (Der2), conservativity, observe that if $I$ is the unit $Set_*$-category having one object $x$ with $hom(x,x) = S^0 = \{1_x, 0\}$, then $D'(I) \simeq D(*)$.  So the family of $Set_*$ functors $I \to A$ for any $Set_*$-category $A$ gives rise a family $*\to \bar{A}$ picking out the nonzero objects, but since any two zero objects are isomorphic, (Der2) for $D$ implies that the resulting family of functors $D'(A) \to D'(I)$ are jointly conservative.

We can also conclude a version of the axiom (Der4) about exact squares from the above theorems about exactness for categories with zeros.  Namely, if
$$\array{ A & \to & B \\ \downarrow & \swArrow & \downarrow \\ C & \to & D }$$
is a comma square of $Set_*$-categories, then
$$\array{ \bar{A} & \to & \bar{B} \\ \downarrow & \swArrow & \downarrow \\ \bar{C} & \to & \bar{D} }$$
is homotopy exact when considered as a square of ordinary categories.  Moreover, the functor $\bar{C} \to \bar{D}$ is always locally null-final.  Thus, by Theorem \ref{HomotopyExactWithZeros1}, our "$Set_*$-enriched derivator" satisfies the Beck-Chevalley condition for any comma square of $Set_*$-categories.

It remains to consider the axiom (Der1) regarding coproducts.  The coproduct "$A\vee B$" of two $Set_*$-categories $A$ and $B$, as $Set_*$-categories, is not quite a "disjoint" union, but rather includes zero morphisms in both directions from each category to the other.  The inclusions $i,j$ of $A$ and $B$ are fully faithful, however, and the induced functors $\bar{i}\colon \bar{A}\to \overline{A\vee B}$ and $\bar{j}\colon \bar{B}\to \overline{A\vee B}$ are locally null-final; hence the left extensions $i_! \colon D'(A) \to D'(A\vee B)$ and $j_!\colon D'(B) \to D'(A\vee B)$ are also fully faithful.

By exactness with zeros, we can show that $j^* i_!$ and $i^* j_!$ send everything to zero.  Therefore, if we define a functor $(i,j)_!\colon D'(A) \times D'(B) \to D'(A\vee B)$ by $(i,j)_!(X,Y) = (i_! X + j_! Y)$, then we have $(i^*, j^*)(i,j)_! \cong Id$.  On the other hand, by fully-faithfulness of $i_!$ and $j_!$, for any $Z\in D'(A\vee B)$ the map $i_! i^* Z \to Z$ is an isomorphism on objects of $A$, and similarly for $j$ and $B$; hence $(i,j)_! (i^*,j^*) \cong Id$ as well.  In particular, $(i^*,j^*)$ is an equivalence of categories, providing the $Set_*$-enriched version of (Der1).

+--{: .query}
What about (Der5)?
=--

Conversely, given a functor $D'$ as above satisfying the axioms as given above, we can define $D(A) = D'(A')$, where $A'$ denotes the free $Set_*$-category on an ordinary category $A$ (i.e. adjoin a new zero morphism between each pair of objects).  This $D$ will automatically satisfy axioms (Der3) (homotopy Kan extensions) and (Der2) (conservativity on objects) since $D'$ does, and it satisfies (Der1) on coproducts since $D'$ does and since $(A+B)' \cong A' + B'$.

+--{: .query}
Does (Der4) on Beck-Chevalley conditions carry back over?  Does this require a "pointed version of Cisinski's theorem"?
=--

Finally, of course each category $D(A) = D'(A')$ has a zero object, so $D$ is a pointed derivator.  Thus, it seems that $Set_*$-enrichment is an essentially equivalent way to express the notion of pointed derivator.  (This approach was used by Franke (see below).)

+--{: .query}
However, what is not immediately clear is that passing from $D$ to $D'$ and back to $D$ again gives the same derivator.  At first sight this seems to require a stronger theorem about homotopy exactness with zeros than is stated above, one in which the square need not be homotopy exact without zeros.
=--

## The pointed reflection

Every derivator can be made pointed in a universal way; given $D$ we define $D_*(A)$ to be the full subcategory of $D(A\times I)$ which is terminal when restricted along $0\colon A\to A\times I$.  It requires a little work to show that this is a derivator.  The main observation being that the inclusion $D_*(A) \hookrightarrow D(A\times I)$ has a left adjoint (the "mapping cone"), which can be constructed just as in the proof of Theorem \ref{RelRefl}.  See III.5 of [Heller's memoir](#HellerHomotopy).

## References

See [[derivator]] for general references.  The pointed reflection is discussed in III.5 of

* [[Alex Heller]], _Homotopy theories_ , Mem. AMS 71:383.
 {#HellerHomotopy}

The definition of relative diagram categories is taken from:

* [[Alex Heller]], _Stable homotopy theories and stabilization_ , [MR](http://www.ams.org/mathscinet-getitem?mr=1431157)
 {#HellerStable}

The $Set_*$-enriched approach is taken as basic in

* Jens Franke, _Uniqueness theorems for certain triangulated categories with an Adams spectral sequence_, [K-theory archive](http://www.math.uiuc.edu/K-theory/0139/)

See also

* [[Moritz Groth]], _Derivators, pointed derivators, and stable derivators_ ([pdf](https://arxiv.org/pdf/1112.3840.pdf))

[[!redirects pointed derivators]]
[[!redirects derivator with a zero object]]
[[!redirects zero object in a derivator]]
[[!redirects derivators with zero objects]]
[[!redirects zero objects in a derivator]]


pointed endofunctor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[endofunctor]] on a [[category]] $\mathcal{A}$ is _pointed_ if it is equipped with a [[natural transformation]] from the [[identity functor]].

## Definition
 {#Definition}

\begin{definition}\label{PointedEndofunctor}
An [[endofunctor]] $S \colon \mathcal{A}\to \mathcal{A}$ is called **pointed** if it is equipped with a [[natural transformation]] $\sigma \colon Id_\mathcal{A} \to S$ from the [[identity functor]] on $\mathcal{A}$.
\end{definition}

\begin{remark}\label{MeaningOfPointedness}
Def. \ref{PointedEndofunctor} is *not* the usual classical notion of a [[pointed object]] in the endo-[[functor category]], since the [[identity functor]] is not in general a [[terminal object]] there. It is however pointed in the sense of a [[pointed object in a monoidal category]], involving a morphism out of the monoidal unit, since the identity functor is the [[tensor unit]] for the canonical [[monoidal category]]-[[structure]] on the [[endofunctor category]] (given by [[horizontal composition]]). In this sense, a pointed endofunctor may be regarded as being equipped with a "monoidal point". A [[monad]] has a canonical such point (see Exp. \ref{Monads} below), usually called the *[[unit of a monad|unit]]*.
\end{remark}

\begin{remark}
The notion of a pointed endofunctors, Def \ref{PointedEndofunctor}, naturally extends to any [[2-category]], where we can define a **pointed endomorphism** to be an endo-[[1-morphism]] $s \colon a \to a$ equipped with a [[2-morphism]] $\sigma \colon id_a \Rightarrow s$ from the [[identity morphism]].
\end{remark}

\begin{definition}
A pointed endofunctor $(S, \sigma)$ (Def. \ref{PointedEndofunctor}) is called **[[well-pointed endofunctor|well-pointed]]** if $S\sigma = \sigma S$ as natural transformations $S \longrightarrow S \circ S$.
\end{definition}

## Properties

### Relation to free algebras

The pointed [[algebras over an endofunctor]] on a pointed endofunctor induce a theory of _[[transfinite construction of free algebras]]_ [[algebra over an endofunctor|over]] general endofunctors.

## Examples

\begin{example}\label{Monads}
A [[monad]] can be regarded as a pointed endofunctor where $\sigma$ is its [[unit of a monad|unit]].
\end{example}

+-- {: .num_prop}
###### Proposition
The pointed endofunctor $(T,\eta)$ underlying a monad $(T : C \to C, \eta: 1\to T, \mu : T^2\to T)$ is well-pointed if and only if $T$ is [[idempotent monad|idempotent]], i.e. $\mu$ is an isomorphism.
=--
+-- {: .proof}
###### Proof
If $\mu$ is an isomorphism then $T\eta=\eta T$ since both are sections of $\mu$.

Conversely, if $T\eta=\eta T$, then $T\eta$ is an inverse for $\mu$.
Indeed,
$$
T\eta_A\circ\mu_A = \mu_{TA}\circ T^2\eta_A = \mu_{TA}\circ T\eta_{TA} = 1_{T^2A},
$$
and $\mu_A\circ T\eta_A = 1_{TA}$ by the right unit law.
=--

## For pointed object in the endofunctor category

Notice that the statement which one might expect, that a pointed endofunctor is a [[pointed object]] in the [[endofunctor category]] is not quite right in general.

The [[terminal object]] of the category of endofunctors on $\mathcal{A}$ is the functor $T$ which sends all objects to $\ast$ and all morphisms to the unique morphism $\ast \to \ast$, where $\ast$ is the terminal object of the category $\mathcal{A}$. So a pointed object in the endofunctor category should be an endofunctor $S:\mathcal{A} \to \mathcal{A}$ equipped with a natural transformation $\sigma:T \to S$.

Rather, a pointed endofunctor is equipped with a map from the [[unit object]] for the monoidal structure on the endofunctor category.

## Related concepts

* [[pointed object]]

* [[well-pointed endofunctor]]

## References

* [[Harvey Wolff]], p. 234 of: *Free monads and the orthogonal subcategory problem*, Journal of Pure and Applied Algebra **13** 3 (1978) 233-242 [doi:10.1016/0022-4049(78)90010-5]

* {#Kelly} [[Max Kelly]], _A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on._  Bull. Austral. Math. Soc. 22 (1980), 1--83. doi:[10.1017/S0004972700006353](http://dx.doi.org/10.1017/S0004972700006353)

[[!redirects pointed endofunctors]]
[[!redirects pointed endomorphism]]
[[!redirects pointed endomorphisms]]


pointed homotopy type

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#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[pointed object|pointed]] [[homotopy type]], i.e. [[pointed object|pointed]] [[infinity-groupoid]], i.e. an [[object]] in the [[coslice (∞,1)-category]] of [[∞Grpd]] under the [[point]]:

$$
  X \in \infty Grpd^{\ast/}
$$

## Related concepts

* [[pointed topological space]], [[pointed simplicial set]]

* [[spectrum]]

## References

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], chapter IV.4 of _[[Calculus of fractions and homotopy theory]]_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967)  ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf))

[[!redirects pointed homotopy types]]
[[!redirects pointed homotopy class]]

[[!redirects pointed infinity-groupoid]]
[[!redirects pointed infinity-groupoids]]

[[!redirects pointed ∞-groupoid]]
[[!redirects pointed ∞-groupoids]]


pointed mapping space

#Contents#
* table of contents
{:toc}

## Idea

The incarnation of  [[internal homs]]/[[mapping spaces]] in the context of [[pointed objects]].

## Definition

+-- {: .num_defn #PointedMappingSpace}
###### Definition

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**

$$
  [X,Y]_*
  \in
  \mathcal{C}^{\ast/}
$$

(the "object of basepoint-preserving maps"), is the [[pullback]]

$$
  \array{
    [X,Y]_* & \overset{}{\longrightarrow} & \ast
    \\
    \downarrow &(pb)& \downarrow
    \\
    [X,Y] & \underset{}{\longrightarrow} & [1,Y]
   }
$$

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 $\ast \otimes X \to \ast \to Y$.

=--

## Properties

+-- {: .num_prop #SmashProductLeftAdjointToPointedMappingSpace}
###### Proposition

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 $X$, and the operation of forming the [[smash product#GeneralSmashProduct|smash product]] with $X$, form a pair of [[adjoint functors]]

$$
  (
  X \wedge (-)
  \;\dashv\;
  [X,-]_\ast
  )
  \;\colon\;
  \mathcal{C}^{\ast/}
  \leftrightarrow
  \mathcal{C}^{\ast/}
  \,.
$$

This makes $\mathcal{C}^{\ast/}$ itself a [[closed monoidal category]], which is [[symmetric monoidal category|symmetric]] if $\mathcal{C}$ is.  The [[tensor unit]] is $I_+$ for $I$ the unit for the monoidal structure on $\mathcal{C}$.

=--

([Elmendorf-Mandell 07, lemma 4.20](#ElmendorfMandell07))

+-- {: .num_remark}
###### Remark

The case when $\mathcal{C}$ is [[cartesian monoidal category|cartesian]], or at least [[semicartesian monoidal category|semicartesian]], is most common in the literature.

=--

+-- {: .num_remark}
###### Remark

If $\mathcal{C}$ is [[monoidal category|monoidal]] but not [[closed monoidal category|closed]], the same definition of the [[smash product#ForGeneralPointedObjects|smash product]] makes $\mathcal{C}^{\ast/}$ [[monoidal category|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 [[nice category of spaces|niceness conditions]] imposed) is not associative.

=--

[[!redirects pointed mapping spaces]]


pointed model category

+-- {: .rightHandSide}
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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[model category]] is __pointed__ if its underlying [[category]]
is a [[pointed category]], i.e., if the unique
[[morphism]] from the [[initial object]] to the [[terminal object]] is an [[isomorphism]], in which case both of them are denoted by $0$ (the [[zero object]]).

In any pointed category, one has a canonical *[[zero morphism]]*
between any [[pair]] of objects $A$ and $B$, given by the [[composition]] $A\to 0\to B$.

The [[homotopy equalizer]] of $f\colon A\to B$ and $0\colon A\to B$ is known as the [[homotopy fiber]] of $f$.

The [[homotopy coequalizer]] of $f\colon A\to B$ and $0\colon A\to B$ is known as the [[homotopy cofiber]] of $f$.

In particuar there is the homotopy (co)-fiber of the [[zero object]] with itself, the [[loop space object]]- and [[reduced suspension]]-operation. Asking these operations to be equivalences in a suitable sense leads to the concept of _[[linear model categories]]_ and _[[stable model categories]]_.

## Examples

Model categories which are pointed without being [[linear model category|linear]] or even [[stable model category|stable]]:

\begin{example}\label{ModelCategoriesOfPointedObjects}
**([[model categories of pointed objects]])**
\linebreak
  Given any model category, its [[model category of pointed objects]] is a pointed model category.

In the case of the [[classical model structure on topological spaces]] this is the [[classical model structure on pointed topological spaces]].
\end{example}

Pointed model categories which are [[stable model category|stable]]:

* [[model structure on spectra]]

## Related entries

* [[homotopy fiber]]

* [[homotopy cofiber]]

* [[pointed category]], [[pointed (∞,1)-category]]

* [[linear model category]]

* [[stable model category]]

## References

* [[Mark Hovey]], Chapter 6 in: _[[Model Categories]]_, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s),   [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

* [Section 4 -- Homotopy fiber sequences](Introduction+to+Homotopy+Theory#HomotopyFiberSequences) in: _[[Introduction to Homotopy Theory]]_

[[!redirects pointed model categories]]


pointed module

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

\tableofcontents

\section{Definition}

\begin{definition}\label{PointedRModule}
Let $R$ be a ([[commutative ring|commutative]]) [[ring]]. Then a **pointed $R$-module** is an $R$-[[module]] $M$ equipped with the further choice of an [[element]] $m \in M$.
A [[homomorphism]] of pointed modules is a homomorphism of the [[underlying]] modules which preserves the choice of the extra points.
\end{definition}

If $R$ is a [[field]], then pointed $R$-modules are also called **pointed $R$-vector spaces**.

If $R$ is the [[integers]] then pointed $R$-modules are also called **[[pointed abelian groups]]**.

\begin{remark}
Equivalently, a pointed $R$-module (Def. \ref{PointedRModule}) is an $R$-[[module]] $M$ equipped with an $R$-[[linear map]] $R \to M$ (which is fixed by its [[image]] of $1 \in R$). Since $R$ is the [[tensor unit]] for the [[tensor product of modules]] in the [[monoidal category]] [[Mod|$R \mathrm{Mod}$]] of $R$-modules and $R$-linear maps, this means that pointed $R$-modules are equivalently [[pointed object in a monoidal category|pointed objects in the monoidal-categorical sense]] (in addition to their [[underlying]] [[sets]] being [[pointed set|pointed objects in the set-theoretic sense]]). Accordingly, the category of pointed modules is equivalently the [[coslice category]] $R \downarrow R \mathrm{Mod}$ of [[Mod|$R \mathrm{Mod}$]] under (the [[underlying]] $R$-module of) $R$.
\end{remark}

\section{Related concepts}

* [[module]]

* [[pointed abelian group]]

* [[pointed object in a monoidal category]]

* [[functorial aspects of the GNS representation]]

[[!redirects pointed module]]
[[!redirects pointed modules]]

[[!redirects category of pointed modules]]

[[!redirects pointed vector space]]
[[!redirects pointed vector spaces]]

[[!redirects category of pointed vector spaces]]


pointed object

> This entry is about the classical notion of *pointed objects* ([[pointed sets]], [[pointed topological spaces]], etc.). Compare the variant notion of *[[pointed object in a monoidal category]]*.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

# Pointed objects
* table of contents
{: toc}

## Idea
 {#Idea}

In a [[category]] $C$ with a [[terminal object]] $\ast \,\in\, C$, a **pointed object** $(X,x)$ is an [[object]] $X$ equipped with a [[global element]], hence with a [[morphism]] of the form $x \,\colon\, \ast \to X$, often called the _[[basepoint]]_.

Pointed objects are distinguished from *[[inhabited objects]] * in that the chosen point is _[[structure]]_ rather than a [[property]].  In particular, a [[homomorphism]] of pointed objects is a [[morphism]] in the original category which preserves the basepoints.  In other words, the category of pointed objects in $C$ is the [[co-slice category]] $\ast/C$ under the [[terminal object]].

{#MoreGenerally} (More generally, one might regard any [[coslice category]] under any object $X \,\in\, C$ as the category of "$X$-pointed objects". This is common in the case where $C$ is a [[monoidal category]] and $X = I$ is its [[tensor unit]], in which case one speaks of *[[pointed objects in a monoidal category]]*. See also [[generalized element]].)

There is an obvious [[forgetful functor]] which forgets the basepoint

$$
  \begin{array}{ccc}
    \ast/C &\longrightarrow& C
    \\
    (X,x) &\mapsto& X
    \mathrlap{\,.}
  \end{array}
$$

If $C$ has [[finite colimit|finite]] [[coproducts]], this functor has a [[left adjoint|left]] [[adjoint functor]] which takes an object $X$ to the coproduct $\ast\sqcup X$, 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]] for this monad.

## Definition

+-- {: .num_defn #SliceCategory}
###### Definition

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 diagram|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 diagram|commuting triangles]] $\array{ && X \\ & \swarrow && \searrow \\ A && \longrightarrow && B }$ in $\mathcal{C}$.

There is the canonical [[forgetful functor]]

$$
  U \;\colon \; \mathcal{C}_{/X}, \mathcal{C}^{X/} \longrightarrow \mathcal{C}
$$

given by forgetting the morphisms to/from $X$.

=--

We here focus on this class of examples:

+-- {: .num_defn #CategoryOfPointedObjects}
###### Definition

For $\mathcal{C}$ a [[category]] with [[terminal object]] $\ast$, the [[coslice category]] (def. \ref{SliceCategory}) $\mathcal{C}^{\ast/}$ 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 diagram|commuting triangles]] of the form

  $$
    \array{
       && \ast
       \\
       & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}}
       \\
       X && \overset{f}{\longrightarrow} && Y
    }
  $$

  (hence morphisms in $\mathcal{C}$ which preserve the chosen points).

=--

## Examples

\begin{example}\label{PointedSets}
 The pointed objects in [[Sets]] are [[pointed sets]].
\end{example}

\begin{example}
  Within the [[doctrine]] of [[cartesian monoidal categories]], all [[internalization|internal notions]] of [[algebra|algebraic]] [[structures]] with [[units]], such as

  * [[natural numbers objects]]

  * [[monoid objects]]

  * [[group objects]]

are in particular (i.e. have [[underlying]]) pointed objects in their ambient categories.
\end{example}

\begin{example}
[[pointed topological spaces|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]].
\end{example}

\begin{example}\label{PointedCategories}
**(relation to pointed categories)**
\linebreak
  If $\mathcal{D}$ is a "[[pointed category]]" in the sense that it contains a [[zero object]] (hence a [[terminal object]] which is also [[initial object|initial]]), then each of its objects carries a unique [[structure]] of a pointed object (by the [[universal property]] of [[initial objects]]).

Moreover, any category $\mathcal{C}^{\ast/}$ of pointed objects is a [[pointed category]], in this sense, with the [[zero object]] of $\mathcal{C}^{\ast/}$ being $\ast \overset{\exists !}{\to} \ast$. This must be the origin of the terminology "pointed category".

Alternatively, it makes (more?) sense to understand under a "pointed category" a [[pointed object]] $(\mathcal{E}, E)$ in [[Categories]], hence a category $\mathcal{E}$ equipped with an [[object]] $E \in \mathcal{E}$. Then one may want to say that the "$E$-pointed objects" $(X,x_E)$ in $(\mathcal{E}, E)$ are morphisms of the form $x_E \colon E \to X$ (i.e. [[generalized elements]] of $X$ at stage $E$).
\end{example}

\begin{example}
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 (Exp. \ref{PointedSets}) is a _0-tuply monoidal 0-category_.
\end{example}

## Properties

### Forgetting and adjoining basepoints

+-- {: .num_defn #BasePointAdjoined}
###### Definition

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. \ref{CategoryOfPointedObjects}, has a [[left adjoint]] given by forming the [[disjoint union]] ([[coproduct]]) with a base point ("adjoining a base point"), this is denoted by

$$
  (-)_+ \coloneqq (-) \sqcup \ast \;\colon \; \mathcal{C} \longrightarrow \mathcal{C}^{\ast/}
  \,.
$$

=--

### Zero object and pointed category structure

+-- {: .num_remark #PointedObjectsHaveZeroObject}
###### Remark

In a [[category of pointed objects]] $\mathcal{C}^{\ast/}$, def. \ref{CategoryOfPointedObjects}, 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]]_

$$
  0
    \;\colon\;
  X \overset{\exists!}{\longrightarrow} 0 \overset{\exists}{\longrightarrow} Y
  \,.
$$

=--

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.

### Limits and colimits

+-- {: .num_prop #LimitsAndColimitsOfPointedObjects}
###### Proposition

Let $\mathcal{C}$ be a [[category]] with all [[limits]] and [[colimits]]. Then also the [[category of pointed objects]] $\mathcal{C}^{\ast/}$, def. \ref{CategoryOfPointedObjects}, has all limits and colimits.

Moreover:

1. 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 $\mathcal{C}$;

1. the colimits are the colimits in $\mathcal{C}$ of the diagrams _with the basepoint adjoined_.

=--

+-- {: .proof}
###### Proof

It is immediate to check the relevant [[universal property]]. For details see at _[slice category -- limits and colimits](overcategory#LimitsAndColimits)_.

=--

### Wedge sum and Smash product
 {#WedgeSumAndSmashProduct}

+-- {: .num_example #WedgeSumAsCoproduct}
###### Example

Given two pointed objects $(X,x)$ and $(Y,y)$, then:

1. their [[product]] in $\mathcal{C}^{\ast/}$ is simply $(X\times Y, (x,y))$;

1. their [[coproduct]] in $\mathcal{C}^{\ast/}$ has to be computed using the second clause in prop. \ref{LimitsAndColimitsOfPointedObjects}: 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:

   $$
     \array{
       \ast &\overset{x}{\longrightarrow}& X
       \\
       {}^{\mathllap{y}}\downarrow &(po)& \downarrow
       \\
       Y &\longrightarrow& X \vee Y
     }
     \,.
   $$

   This is called the _[[wedge sum]]_ operation on pointed objects.

Generally for a set $\{X_i\}_{i \in I}$ in $\mathcal{C}^{\ast/}$

1. their [[product]] is formed in $Top$, with the new basepoint canonically induced;

1. their [[coproduct]] is formed by the [[colimit]] in $Top$ over the diagram with a basepoint adjoined, and is called the [[wedge sum]] $\vee_{i \in I} X_i$.

=--

+-- {: .num_example}
###### Example

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. \ref{WedgeSumAsCoproduct}, of $n$-spheres, one for each $n$-cell of $X$:

$$
  X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n
  \,.
$$

=--

+-- {: .num_defn #SmashProductOfPointedObjects}
###### Definition

For $\mathcal{C}^{\ast/}$ a [[category of pointed objects]] with [[finite limits]] and [[finite colimits]], the _[[smash product]]_ is the [[functor]]

$$
  (-)\wedge(-)
  \;\colon\;
  \mathcal{C}^{\ast/}
  \times
  \mathcal{C}^{\ast/}
  \longrightarrow
  \mathcal{C}^{\ast/}
$$

given by

$$
  X \wedge Y
  \;\coloneqq\;
  \ast
  \underset{X\sqcup Y}{\sqcup} (X \times Y)
  \,,
$$

hence by the [[pushout]] in $\mathcal{C}$

$$
  \array{
    X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y
    \\
    \downarrow && \downarrow
    \\
    \ast &\longrightarrow& X \wedge Y
  }
  \,.
$$

In terms of the [[wedge sum]] from def. \ref{WedgeSumAsCoproduct}, this may be written concisely as

$$
  X \wedge Y = \frac{X\times Y}{X \vee Y}
  \,.
$$

=--

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 $\mathcal{C}^{\ast/}$|

+-- {: .num_example #WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces}
###### Example

For $X, Y \in Top$, with $X_+,Y_+ \in Top^{\ast/}$, def. \ref{BasePointAdjoined}, then

* $X_+ \vee Y_+ \simeq (X \sqcup Y)_+$;

* $X_+ \wedge Y_+ \simeq (X \times Y)_+$.

=--

+-- {: .proof}
###### Proof

By example \ref{WedgeSumAsCoproduct}, $X_+ \vee Y_+$ is given by the colimit in $Top$ over the diagram

$$
  \array{
    && && \ast
    \\
    && & \swarrow && \searrow
    \\
    X &\,\,& \ast && && \ast &\,\,& Y
  }
  \,.
$$

This is clearly $A \sqcup \ast \sqcup B$. Then, by definition \ref{SmashProductOfPointedObjects}

$$
  \begin{aligned}
    X_+ \wedge Y_+
    & \simeq
    \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)}
    \\
    & \simeq
    \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast}
    \\
    & \simeq
    X \times Y \sqcup \ast
    \,.
  \end{aligned}
$$

=--

+-- {: .num_example #StandardReducedCyclinderInTop}
###### Example

Let $\mathcal{C}^{\ast/} = Top^{\ast/}$ be [[pointed topological spaces]]. Then

$$
  I_+ \in Top^{\ast/}
$$

denotes the standard interval object $I = [0,1]$, with a disjoint basepoint adjoined, def. \ref{BasePointAdjoined}. Now for $X$ any [[pointed topological space]], then

$$
  X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)
$$

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](#MappingCones).)

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 \ref{WedgeSumAsCoproduct}:

$$
  X \vee X \longrightarrow X \wedge (I_+)
  \,.
$$

=--

### Fibers and cofibers -- kernels and cokernels

+-- {: .num_defn #FiberAndCofiberInPointedObjects}
###### Definition

Given a morphism $f \colon X \longrightarrow Y$ in a [[category of pointed objects]] $\mathcal{C}^{\ast/}$, def. \ref{CategoryOfPointedObjects}, with finite limits and colimits,

1. its **[[fiber]]** or **[[kernel]]** is the [[pullback]] of the point inclusion

   $$
     \array{
       fib(f) &\longrightarrow& X
       \\
       \downarrow &(pb)& \downarrow^{\mathrlap{f}}
       \\
       \ast &\longrightarrow& Y
     }
   $$

1. its **[[cofiber]]** or **[[cokernel]]** is the [[pushout]] of the point projection

   $$
     \array{
       X &\overset{f}{\longrightarrow}& Y
       \\
       \downarrow &(po)& \downarrow
       \\
       \ast &\longrightarrow& cofib(f)
     }
     \,.
   $$

=--

+-- {: .num_remark }
###### Remark

In the situation of def. \ref{FiberAndCofiberInPointedObjects}, both the pullback as well as the pushout are equivalently computed in $\mathcal{C}$. For the pullback this is the first clause of prop. \ref{LimitsAndColimitsOfPointedObjects}. 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

$$
  \array{
    \ast
    \\
    & \searrow
    \\
    & & X &\overset{f}{\longrightarrow}& Y
    \\
    & & \downarrow &&
    \\
    & & \ast &&
   }
$$

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]].)

=--

### Closed and monoidal structure
 {#ClosedMonoidalStructure}

+-- {: .num_defn #PointedMappingSpace}
###### Definition

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]]**

$$
  [X,Y]_*
  \in
  \mathcal{C}^{\ast/}
$$

(the "object of basepoint-preserving maps"), is the [[pullback]]

$$
  \array{
    [X,Y]_* & \overset{}{\longrightarrow} & \ast
    \\
    \downarrow &(pb)& \downarrow
    \\
    [X,Y] & \underset{}{\longrightarrow} & [1,Y]
   }
$$

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 $\ast \otimes X \to \ast \to Y$.

=--

+-- {: .num_prop #SmashProductLeftAdjointToPointedMappingSpace}
###### Proposition

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. \ref{PointedMappingSpace}, and the operation of forming the [[smash product]] with $X$, def. \ref{SmashProductOfPointedObjects} form a pair of [[adjoint functors]]

$$
  (
  X \wedge (-)
  \;\dashv\;
  [X,-]_\ast
  )
  \;\colon\;
  \mathcal{C}^{\ast/}
  \leftrightarrow
  \mathcal{C}^{\ast/}
  \,.
$$

This makes $\mathcal{C}^{\ast/}$ itself a [[closed monoidal category]], which is [[symmetric monoidal category|symmetric]] if $\mathcal{C}$ is.  The [[tensor unit]] is $I_+$ (def. \ref{BasePointAdjoined}) for $I$ the unit for the monoidal structure on $\mathcal{C}$.

=--

([Elmendorf-Mandell 07, lemma 4.20](#ElmendorfMandell07))

+-- {: .num_remark}
###### Remark

The case when $\mathcal{C}$ is [[cartesian monoidal category|cartesian]], or at least [[semicartesian monoidal category|semicartesian]], is most common in the literature.

=--

+-- {: .num_remark}
###### Remark

If $\mathcal{C}$ is [[monoidal category|monoidal]] but not [[closed monoidal category|closed]], the same definition \ref{SmashProductOfPointedObjects} of the [[smash product]] makes $\mathcal{C}^{\ast/}$ [[monoidal category|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 [[nice category of spaces|niceness conditions]] imposed) is not associative.

=--

For [[base change]] functoriality of these structures see at _[Wirthmüller context -- Examples -- On pointed objects](Wirthmüller+context#PointedObjects)_.

### Monadicity
 {#Monadicity}

Pointed objects are the [[algebras over a monad]] of the [[monad]] $X \mapsto X \sqcup \ast$ (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](accessible+monad#CategoryOfAlgebras)_.

### Classifying topos

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.

### Model category structure
 {#ModelCategoryStructure}

+-- {: .num_prop #ModelStructureOnSliceCategory}
###### Proposition
**([[model structure on pointed objects]])**

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. \ref{SliceCategory}, carry model structures themselves -- the **[[model structure on a slice category|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. \ref{CategoryOfPointedObjects}, in a model category $\mathcal{C}$ becomes itself a model category this way.

=--

+-- {: .proof}
###### Proof

The model structure as claimed is immediate by inspection.

=--

+-- {: .num_example #ClassicalPointedHomotopyCategory}
###### Example

For $\mathcal{C} = Top_{Quillen}$, the [[classical model structure on topological spaces]], then the model structure on [[pointed topological spaces]] induced via prop. \ref{ModelStructureOnSliceCategory} 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_ $Ho(Top^{\ast/})$.

=--

+-- {: .num_example #NonDegenerateBasepointAsCofibrantObjects}
###### Example

The fibrant objects in the pointed model structure $\mathcal{C}^{\ast/}$, prop. \ref{ModelStructureOnSliceCategory}, 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.

=--

+-- {: .num_example #HomotopyCategoryOfPointedModelStructureIsEnrichedInPointedSets}
###### Example

For $\mathcal{C}$ any [[model category]], with $\mathcal{C}^{\ast/}$ its [[slice model structure|pointed model structure]] according to prop. \ref{ModelStructureOnSliceCategory}, then the corresponding [[homotopy category of a model category|homotopy category]] is, by remark \ref{PointedObjectsHaveZeroObject}, canonically [[enriched category|enriched]] in [[pointed sets]], in that its [[hom-functor]] is of the form

$$
  [-,-]_\ast
  \;\colon\;
  Ho(\mathcal{C}^{\ast/})^\op
  \times
  Ho(\mathcal{C}^{\ast/})
  \longrightarrow
  Set^{\ast/}
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

If $\mathcal{C}$ is a [[monoidal model category]] with cofibrant [[tensor unit]], then the pointed model structure on  $\mathcal{C}^{\ast/}$ (prop. \ref{ModelStructureOnSliceCategory}) is also a [[monoidal model category]], and the [[smash product]]$\dashv$[[mapping space]] adjunction of prop. \ref{SmashProductLeftAdjointToPointedMappingSpace} is a [[Quillen adjunction]]

$$
  (
  X \wedge (-)
  \;\dashv\;
  (-)^X
  )
  \;\colon\;
  \mathcal{C}^{\ast/}
  \leftrightarrow
  \mathcal{C}^{\ast/}
  \,.
$$

=--

## Related concepts

* [[bi-pointed object]]

* [[pointed type]]

* [[smash product]]

* [[topos of pointed objects]]

* [[fibration of points]]

* [[model structure on pointed objects]]

## References

* [[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](http://arxiv.org/abs/0710.0082))
 {#ElmendorfMandell07}

[[!redirects pointed object]]
[[!redirects pointed objects]]

[[!redirects base point]]
[[!redirects base points]]
[[!redirects basepoint]]
[[!redirects basepoints]]

[[!redirects category of pointed objects]]
[[!redirects categories of pointed objects]]


pointed object in a monoidal category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

\tableofcontents

## Idea

Just as the classical notion of *[[pointed objects]]* refers to *morphisms* whose [[domain]] is [[terminal object|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.

## Examples

\begin{example}
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.
\end{example}

\begin{example}
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]].
\end{example}

\begin{example}
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]].
\end{example}

\begin{example}
A [[bi-pointed object|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.
\end{example}

\begin{example}
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 functor|lax monoidal]] change of base functor
$$U = V(I, -): V \to Set$$
and a pointing of an object $v$ of $V$ is equivalent to a pointing of its underlying object in the classical sense, $1 \to U(v)$.
\end{example}

\begin{example}
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 [[affine space#ClosedMonoidal|closed monoidal]] structure of affine spaces.
\end{example}

## References

The term appears, for instance, in:

* {#Melliès09} [[Paul-André Melliès]], Nicolas Tabareau & Christine Tasson, p. 4 of: *An Explicit Formula for the Free Exponential Modality of Linear Logic*, in: *Automata, Languages and Programming. ICALP 2009*, Lecture Notes in Computer Science, **5556**, Springer (2009) [[doi:10.1007/978-3-642-02930-1_21](https://doi.org/10.1007/978-3-642-02930-1_21)]

[[!redirects pointed objects in a monoidal category]]
[[!redirects pointed objects in monoidal categories]]


pointed set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A __pointed set__ is a [[pointed object]] in [[Set]], hence a [[set]] $S$ equipped with a chosen element $s$ of $S$. (Compare [[inhabited set]], where the element is not specified.)

Since we can identify a (set-theoretic) element of $S$ with a (category-theoretic) [[global element]] (a morphism $s: {\ast} \to S$ from the [[terminal object]] ${\ast}$), we see that a pointed set is an object of the [[under category]] $\pt \downarrow \Set$, or coslice category ${\ast}/\Set$, of objects under the [[singleton set]] ${\ast}$.

## The category of pointed sets

### Definition

+-- {: .un_defn}
###### Definition

The [[category]] $\Set_{\ast}$ of pointed sets is the [[under category]] ${\ast}/\Set$ of [[Set]] under the [[singleton set]] ${\ast}$.

=--

So a morphism $(S_1, s_1) \to (S_2, s_2)$ is a map between sets which maps these chosen elements to each other, i.e., commuting triangles

$$
  \array{
    && {\ast}
    \\
    & ^{s_1}\swarrow && \searrow^{s_2}
    \\
    S_1
    &&\to&&
    S_2
  }
 \,.
$$

The category $\Set_{\ast}$ naturally comes with a [[forgetful functor]] $p : \Set_{\ast} \to \Set$ which forgets the tip of these triangles.

### Properties

+-- {: .un_prop}
###### Proposition

Equipped with the [[smash product]] ${\otimes} := {\wedge}$ of pointed sets, $(\Set_{\ast}, {\wedge})$ is a [[closed monoidal category|closed]] [[symmetric monoidal category]].

The [[internal hom]] $\Set_{\ast}(X,Y)$ is the [[hom-set]] in ${\ast}/\Set$ pointed by the morphism $X \to Y$ that sends everything to the basepoint in $Y$.

=--

See at _[[pointed object]]_ for more details.

#### Pointed objects in the category of pointed sets

The [[tensor unit]] of pointed sets is the [[boolean domain]] $\mathbb{2}$, and [[pointed object in a monoidal category|pointed objects]] in the category of pointed sets are pointed sets $A$ with a point-preserving function $\mathbb{2} \to A$.

#### Natural numbers object

In [[classical mathematics]], the [[natural numbers object]] in $Set_*$ is the [[set]] of [[extended natural numbers]] $\overline{\mathbb{N}} = \mathbb{N} + \{\infty\}$, and comes with point-preserving functions $z_0:\mathbb{2} \to \overline{\mathbb{N}}$ and $z_s:\overline{\mathbb{N}} \to \overline{\mathbb{N}}$ such that for all pointed sets $A$ and point-preserving functions $f:\mathbb{2} \to A$, $g: A \to A$, there is a unique point-preserving function $\phi_{f, g}:\overline{\mathbb{N}} \to A$ making the following diagram commute:

$$\array{
\mathbb{2} & \stackrel{z_0}{\to} & \overline{\mathbb{N}} & \stackrel{z_s}{\leftarrow} & \overline{\mathbb{N}} \\
 & \mathllap{f} \searrow & \downarrow \mathrlap{\phi_{f, g}} & & \downarrow \mathrlap{\phi_{f, g}} \\
 & & A & \underset{g}{\leftarrow} & A
}$$

The point-preserving function $z_0$ represents the function which takes the [[boolean]] [[true]] to $\infty$ and [[false]] to zero, and $z_s$ represents the point-preserving function which takes a natural number to its successor and $\infty$ to $\infty$.

The [[absorption monoid]] structure on $\overline{\mathbb{N}}$ is defined by double [[induction]] on $\overline{\mathbb{N}}$, we define

$$(-)+(-):\overline{\mathbb{N}} \times \overline{\mathbb{N}} \to \overline{\mathbb{N}} \wedge \overline{\mathbb{N}} \to \overline{\mathbb{N}}$$

by

$$z_0(p) + z_0(q) = z_0(p \vee q) \qquad z_s(m) + z_0(q) = z_s(m + z_0(q))$$
$$z_0(p) + z_s(n) = z_s(z_0(p) + n) \qquad z_s(m) + z_s(n) = z_s(z_s(m + n))$$

for all $p, q \in \mathbb{2}$ and $m, n \in \overline{\mathbb{N}}$, where $p \vee q$ is the [[disjunction]] of [[booleans]] $p$ and $q$ (recall the definition of [[addition]] in the [[natural numbers]], inductively defined by $0(p) + 0(q) = 0(p \cdot q)$, $s(m) + 0(p) = s(m + 0(p))$, $0(p) + s(n) = s(0(p) + n)$, and $s(m) + s(n) = s(s(m + n))$ for all $p, q \in \mathbb{1}$ and $m, n \in \mathbb{N}$). It is a [[commutative monoid]] and represents addition in $\overline{\mathbb{N}}$.

In [[constructive mathematics]], the extended natural numbers $\overline{\mathbb{N}}$ and the [[disjoint union]] $\mathbb{N} + \{\infty\}$ are no longer the same; it is $\mathbb{N} + \{\infty\}$ which remains the natural numbers object in $Set_*$.

#### Interpretation as universal Set-bundle
 {#InterpretationAsUniversalSetBundle}

The morphism $\Set_{\ast} \to \Set$ is an example of a
[[generalized universal bundle]]: the _universal [[Set]]-bundle_.
The entire structure here can be understood as arising from the (strict) pullback diagram

$$
  \array{
    \Set_{\ast} &\to& \pt
    \\
    \downarrow && \downarrow^{\pt \mapsto {\ast}}
    \\
    [I,\Set]
    &\stackrel{d_0}{\to}& \Set
    \\
    \downarrow^{d_1}
    \\
    \Set
  }
$$

in the 1-category [[Cat]], where

* $I = \{0 \to 1\}$ is the [[interval category]];

* $[I, \Set] = Arr(\Set)$ is the [[closed monoidal category|internal hom]] category which here is the [[arrow category]] of $\Set$;

* $d_i := [j_i, \Set]$ are the images of the two injections
$j_i : \pt \to I$ of the point to the left and the right end of the interval, respectively --- so these functors evaluate on the left and right end of the interval, respectively;

* the square is a pullback;

* the total vertical functor is the forgetful functor $p : \Set_{\ast} \to \Set$.

The way in which $\Set_{\ast} \to \Set$ is the "universal Set-bundle" is discussed pretty explicitly in

* Kathryn Hess, _[[HessLackBundCat.pdf:file]]_ .

(The discussion there becomes more manifestly one of bundles if one regards all morphisms $C \to \Set$ appearing there as being the right legs of [[anafunctor|anafunctors]]. )

#### Interpretation as 2-subobject-classfier

Observing that usual morphism into the [[subobject classifier]] $\Omega$ of the [[topos]] [[Set]] is the
[[universal truth-value bundle]]
$\{\top\} \to \TV$, and noticing that $TV = (-1)Cat$ and $Set = 0Cat$ suggests that $Set_* \to Set$ is a [[vertical categorification|categorified]] subobject classifier:
indeed, it is the subobject classifier in the [[2-topos]] [[Cat]].

For discussion of this point see

* David Corfield: _101 things to do with a 2-classifier_ ([blog](http://golem.ph.utexas.edu/category/2008/01/101_things_to_do_with_a_2class.html))

It was David Roberts who pointed out in

* David Roberts, [comment to: 101 things to do with a 2-classifier](http://golem.ph.utexas.edu/category/2008/01/101_things_to_do_with_a_2class.html#c014559)

the relation between these higher classifiers and higher [[generalized universal bundle|generalized universal bundles]], motivated by the observations on principal universal 1- and 2-bundles in

* David Roberts, Urs Schreiber, _The inner automorphism 3-group of a strict 2-group_, Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244, ([arXiv](http://arxiv.org/abs/0708.1741v2)).

## Related concepts

* [[pointed object]]

* [[pointed topological space]]

* [[pointed simplicial set]]

* [[pointed homotopy type]]

[[!redirects pointed sets]]

[[!redirects category of pointed sets]]


pointed simplicial set

#Contents#
* table of contents
{:toc}

## Idea

A [[pointed object|pointed]] [[simplicial set]] $X \in sSet^{\ast/}$.

## Related concepts

* [[reduced simplicial set]]

* [[pointed topological space]]

* [[loop space]], [[reduced suspension]]

* [[smash product]], [[wedge sum]]

* [[pointed homotopy type]]

* [[spectrum]]

* [[classical model structure on pointed topological spaces]]

[[!redirects pointed simplicial sets]]

[[!redirects based simplicial set]]
[[!redirects based simplicial sets]]


pointed space

Concretely, by a _pointed space_ or _based space_ is typically meant a _[[pointed topological space]]_. See there for more.

Generally, a _pointed space_, or __based space__, is a [[pointed object]] in a [[category]] of [[spaces]] of sorts.

That is, a pointed space is a space $X$ together with a point $a$ of $X$, called the __basepoint__, or __base point__, of the pointed space.  Here 'point' in general means [[global element]]; this matches the usual meaning of 'point' in all of the common categories of spaces, even non-[[concrete category|concrete]] ones like the category of [[locale]]s.

Many constructions in [[homotopy theory]], such as the [[fundamental group]] and other [[homotopy groups]], are defined on [[pointed topological spaces]].  One may remove the dependence on a basepoint by passing to the [[fundamental groupoid]].

[[!redirects based space]]
[[!redirects pointed spaces]]
[[!redirects based spaces]]


pointed topological space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _pointed topological space_ (often _[[pointed space]]_, for short) is a [[topological space]] equipped with a choice of one of its [[points]] ([[elements]]). If the inclusion of that point is a [[Hurewicz cofibration]] then one speaks of a *[[well-pointed topological space]]*.

Although  this concept may seem simple, pointed topological spaces play a central role for instance in [[algebraic topology]] as domains for  [[reduced cohomology|reduced]] [[generalized (Eilenberg-Steenrod) cohomology theories]] and as an ingredient for the definition of [[spectra]].

One reason why pointed topological spaces are important is that the [[category]] which they form is an intermediate stage in the [[stabilization]] of [[homotopy theory]] (the [[classical model structure on topological spaces|classical homotopy theory of topological spaces]]) to [[stable homotopy theory]]:

The category of pointed topological spaces has a [[zero object]] (the [[point space]] itself) and the canonical [[tensor product]] on pointed spaces is the _[[smash product]]_, which is non-[[cartesian monoidal category]], in contrast to the plain [[product space|product of topological space]].

## Definition

A _pointed topological space_ is a [[topological space]] $(X,\tau)$ equipped with a choice of point $x \in X$.
A [[homomorphism]] between pointed topological space $(X,x)$ $(Y,y)$ is a [[continuous function]] $f \colon X \to Y$ which preserves the chosen basepoints in that $f(x) = y$.

### The category of pointed topological spaces

Stated in the language of [[category theory]], this means that pointed topological spaces are the _[[pointed objects]]_ in the [[category]] [[Top]] of topological spaces. This is the [[coslice category]] $Top^{\ast/}$ of topological spaces "under" the [[point space]] $\ast$:

an [[object]] in $Top^{\ast/}$ is equivalently a [[continuous function]] $x \colon \ast  \to (X,\tau)$, which is equivalently just a choice of point in $X$, and a [[morphism]] in $Top^{\ast/}$ is a morphism $f \colon X \to Y$ in [[Top]] (hence a continuous function), such that this triangle [[diagram]] [[commuting diagram|commutes]]

$$
  \array{
    && \ast
    \\
    & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}}
    &&
    \\
    X && \underset{f}{\longrightarrow} && Y
  }
$$

which equivalently means that $f(x) = y$.

### Forgetting and adjoining basepoints
 {#ForgettingAndAdjoiningBasepoints}

+-- {: .num_defn #BasePointAdjoined}
###### Definition

The [[forgetful functor]] $Top^{\ast/} \to Top$ has a [[left adjoint]] given by forming the [[disjoint union space]] ([[coproduct]] in [[Top]]) with a [[point space]] ("adjoining a base point"), this is denoted by

$$
  (-)_+ \coloneqq (-) \sqcup \ast \;\colon \; Top \longrightarrow Top^{\ast/}
  \,.
$$

=--

### Wedge sum and Smash product

+-- {: .num_example #WedgeSumAsCoproduct}
###### Example

Given two pointed topological spaces $(X,x)$ and $(Y,y)$, then:

1. their [[Cartesian product]] in $Top^{\ast/}$ is simply their [[product topological space]] $X \times Y$ equipped with the [[pair]] of basepoints $(X\times Y, (x,y))$;

1. their [[coproduct]] in $Top^{\ast/}$ has to be computed using the second clause in [this prop.](pointed+object#LimitsAndColimitsOfPointedObjects): since the point $\ast$ has to be adjoined to the diagram, it is given not by the coproduct in $Top$ (which is the [[disjoint union space]]), but by the [[pushout]] in $Top$ of the form:

   $$
     \array{
       \ast &\overset{x}{\longrightarrow}& X
       \\
       {}^{\mathllap{y}}\downarrow
         &\mathclap{{}^{{}_{(po)}}}&
       \downarrow
       \\
       Y &\longrightarrow& X \vee Y
     }
     \,.
   $$

   This is called the _[[wedge sum]]_ operation on pointed objects.

   This is the [[quotient topological space]] of the [[disjoint union space]] under the [[equivalence relation]] which identifies the two basepoints:

   $$
     X \vee Y \;\simeq\; (X \sqcup Y)/(x \sim y)
   $$

Generally for a set $\{(X_i,x_i)\}_{i \in I}$ of pointed topological spaces

1. their [[product]] is formed in [[Top]], as the [[product topological space]] with the [[Tychonoff topology]], with the [[tuple]] $(x_i)_{i \in I} \in \underset{i \in I}{\prod} X_i$ of basepoints being the new basepoint;

1. their [[coproduct]] is formed by the [[colimit]] in $Top$ over the [[diagram]] with a basepoint adjoined, and is called the [[wedge sum]] $\vee_{i \in I} X_i$, which is the [[quotient topological space]] of the [[disjoint union space]] with all the basepoints identified:

   $$
     \underset{i \in I}{\vee} X_i
     \;\simeq\;
     \left(\underset{i \in I}{\sqcup} X_i\right)/(x_i \sim x_j)_{i,j \in I}
    \,.
   $$

=--

+-- {: .num_example}
###### Example

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. \ref{WedgeSumAsCoproduct}, of $n$-spheres, one for each $n$-cell of $X$:

$$
  X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n
  \,.
$$

=--

+-- {: .num_defn #SmashProductOfPointedObjects}
###### Definition

The _[[smash product]]_ of pointed topological spaces is the [[functor]]

$$
  (-)\wedge(-)
    \;\colon\;
  Top^{\ast/}
    \times
  Top^{\ast/}
    \longrightarrow
  Top^{\ast/}
$$

given by

$$
  X \wedge Y
  \;\coloneqq\;
  \ast
  \underset{X\sqcup Y}{\sqcup} (X \times Y)
  \,,
$$

hence by the [[pushout]] in $Top$ of he frm

$$
  \array{
    X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& X \wedge Y
  }
  \,.
$$

In terms of the [[wedge sum]] from def. \ref{WedgeSumAsCoproduct}, this may be written concisely as the [[quotient space]] ([this def](quotient+space#QuotientBySubspace)) of the [[product topological space]] by the [[subspace]] constituted by the  [[wedge sum]]

$$
  X \wedge Y \simeq \frac{X\times Y}{X \vee Y}
  \,.
$$

=--
t
$\,$

| symbol | name | category theory |
|--------|------|-----------------|
| $X \times Y$ | [[product space]] | [[product]] in $Top^{\ast/}$ |
| $X \vee Y$ | [[wedge sum]] | [[coproduct]] in $Top^{\ast/}$ |
| $X \wedge Y = \frac{X \times Y}{X \vee Y}$ | [[smash product]] | [[tensor product]] in $Top^{\ast/}$ |

+-- {: .num_example #WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces}
###### Example

For $X, Y \in Top$, with $X_+,Y_+ \in Top^{\ast/}$, def. \ref{BasePointAdjoined}, then

* $X_+ \vee Y_+ \simeq (X \sqcup Y)_+$;

* $X_+ \wedge Y_+ \simeq (X \times Y)_+$.

=--

+-- {: .proof}
###### Proof

By example \ref{WedgeSumAsCoproduct}, $X_+ \vee Y_+$ is given by the colimit in $Top$ over the diagram

$$
  \array{
    && && \ast
    \\
    && & \swarrow && \searrow
    \\
    X &\,\,& \ast && && \ast &\,\,& Y
  }
  \,.
$$

This is clearly $A \sqcup \ast \sqcup B$. Then, by definition \ref{SmashProductOfPointedObjects}

$$
  \begin{aligned}
    X_+ \wedge Y_+
    & \simeq
    \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)}
    \\
    & \simeq
    \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast}
    \\
    & \simeq
    X \times Y \sqcup \ast
    \,.
  \end{aligned}
$$

=--

+-- {: .num_example #StandardReducedCyclinderInTop}
###### Example

Let $I \coloneqq [0,1] \subset \mathbb{R}$ be the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]].

Hence

$$
  I_+ \in Top^{\ast/}
$$

is the interval with a disjoint basepoint adjoined, def. \ref{BasePointAdjoined}.

Now for $X$ any [[pointed topological space]], then the [[smash product]] (def. \ref{SmashProductOfPointedObjects})

$$
  X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)
$$

is the **[[reduced cylinder]]** over $X$: the result of forming the ordinary [[cylinder]] over $X$, and then identifying the interval over the basepoint of $X$ with the point.

(Generally, any construction in $Top$ properly adapted to pointed spaces is called the "reduced" version of the unpointed construction. Notably so for "[[reduced suspension]]" which we come to [below](#MappingCones).)

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 \ref{WedgeSumAsCoproduct}:

$$
  X \vee X \longrightarrow X \wedge (I_+)
  \,.
$$

=--

### Mapping (co-)cones

Recall that the _[[cone]]_ on a [[topological space]] $X$ is the [[quotient space]] of the [[product space]] with the closed interval

$$
  Cone(X)
   =
  (X \times [0,1])/( X \times \{0\} )
  \,.
$$

If $X$ is pointed with basepoint $x \in X$, then the  _reduced cone_ is the further quotient by the copy of the interval over the basepoint

$$
  Cone(X,x) = Cone(X) / ( \{x\} \times [0,1] )
  \,.
$$

For $f \colon X \to Y$ a [[continuous function]], then

1. the _[[mapping cylinder]]_ of $f$ is the [[attachment space]]

   $$
     Cyl(f) \coloneqq Y \cup_f Cyl(X)
   $$

1. the _[[mapping cone]]_ of $f$ is the [[attachment space]]

   $$
     Cone(f) \coloneqq Y \cup_f Cone(X)
   $$

accordingly if $f \colon X \to Y$ is a continuous function between pointed spaces which preserves the basepoint, then the analogous construction with the [[reduced cylinder]] and the reduce cone, respectively, yield the _reduced mapping cyclinder_ and the _reduced mapping cone_.

We now say this again in terms of [[pushouts]]:

+-- {: .num_defn #MappingConeAndMappingCocone}
###### Definition

For $f \colon X \longrightarrow Y$ a [[continuous function]] between pointed spces, its **reduced [[mapping cone]]** is the space

$$
  Cone(f)
    \coloneqq
  \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y
$$

in the [[colimit|colimiting]] [[diagram]]

$$
  \array{
    && X &\stackrel{f}{\longrightarrow}& Y
    \\
    && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}}
    \\
    X &\stackrel{i_0}{\longrightarrow}& Cyl(X)
    \\
    \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow
    \\
    {*} &\longrightarrow& &\longrightarrow& Cone(f)
  }
  \,,
$$

where $Cyl(X)$ is the  [[reduced cylinder]] from def. \ref{StandardReducedCyclinderInTop}.

=--

+-- {: .num_prop #ConeAndMappingCylinder}
###### Proposition

The colimit appearing in the definition of the reduced [[mapping cone]] in def. \ref{MappingConeAndMappingCocone} is equivalent to three consecutive [[pushouts]]:

$$
  \array{
    && X &\stackrel{f}{\longrightarrow}& Y
    \\
    && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}}
    \\
    X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f)
    \\
    \downarrow &(po)& \downarrow & (po) & \downarrow
    \\
    {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f)
  }
  \,.
$$

The two intermediate objects appearing here are called

* the plain **reduced [[cone]]**  $Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X)$;

* the **reduced [[mapping cylinder]]** $Cyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y$.

=--

+-- {: .num_defn #SuspensionAndLoopSpaceObject}
###### Definition

Let $X \in Top^{\ast/}$ be any pointed topological space.

The [[mapping cone]], def. \ref{ConeAndMappingCylinder}, of $X \to \ast$ is called the  **[[reduced suspension|reduced]] [[suspension]]** of $X$, denoted

   $$
     \Sigma X = Cone(X\to\ast)\,.
   $$

   Via prop. \ref{ConeAndMappingCylinder} this is equivalently the coproduct of two copies of the cone on $X$ over their base:

   $$
     \array{
       && X &\stackrel{}{\longrightarrow}& \ast
       \\
       && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}}
       \\
       X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X)
       \\
       \downarrow &(po)& \downarrow & (po) & \downarrow
       \\
       {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X
     }
     \,.
   $$

   This is also equivalently the [[cofiber]]f of $(i_0,i_1)$, hence (example \ref{WedgeSumAsCoproduct}) of the [[wedge sum]] inclusion:

   $$
     X \vee X
      =
     X \sqcup X
      \overset{(i_0,i_1)}{\longrightarrow}
     Cyl(X)
       \overset{cofib(i_0,i_1)}{\longrightarrow}
     \Sigma X
     \,.
   $$

=--

+-- {: .num_prop #ReducedSuspensionBySmashProductWithCircle}
###### Proposition

The [[reduced suspension]] objects (def. \ref{SuspensionAndLoopSpaceObject}) induced from the standard [[reduced cylinder]] $(-)\wedge (I_+)$ of example \ref{StandardReducedCyclinderInTop} are isomorphic to the [[smash product]] (def. \ref{SmashProductOfPointedObjects}) with the [[circle] (the [[1-sphere]])

  $$
    cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X
    \,,
  $$

=--

+-- {: .num_prop #UnreducedMappingConeAsReducedConeOfBasedPointAdjoined}
###### Proposition

For $f \colon X \longrightarrow Y$ a morphism in [[Top]], then its unreduced mapping cone with respect to the standard cylinder object $X \times I$ def. \ref{TopologicalInterval}, is isomorphic to the reduced mapping cone, of the morphism $f_+ \colon X_+ \to Y_+$ (with a basepoint adjoined) with respect to the standard [[reduced cylinder]]:

$$
  Cone'(f) \simeq Cone(f_+)
  \,.
$$

=--

+-- {: .proof}
###### Proof

By example \ref{WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces}, $Cone(f_+)$ is given by the colimit in $Top$ over the following diagram:

$$
  \array{
    \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast
    \\
    \downarrow && && \downarrow
    \\
    \ast &\longrightarrow& &\longrightarrow& Cone(f_+)
  }
  \,.
$$

We may factor the vertical maps to give

$$
  \array{
    \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast
    \\
    \downarrow && && \downarrow
    \\
    \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+
    \\
    \downarrow && && \downarrow
    \\
    \ast &\longrightarrow& &\longrightarrow& Cone'(f)
  }
  \,.
$$

This way the top part of the diagram (using the [[pasting law]] to compute the colimit in two stages) is manifestly a cocone under the result of applying $(-)_+$ to the diagram for the unreduced cone. Since $(-)_+$ is itself given by a colimit, it preserves colimits, and hence gives the partial colimit $Cone'(f)_+$ as shown. The remaining pushout then contracts the remaining copy of the point away.

=--

## Properties

### General

Most of the relevant constructions on pointed topological spaces are immediate specializations of the general construction discussed at _[[pointed object]]_.

### Relation to one-point compactification

+-- {: .num_prop #OnePointCompactificationAndSmashProduct}
###### Proposition
**([[one-point compactification intertwines Cartesian product with smash product]])

On the [[subcategory]] $Top_{LCHaus}$ of [[Top]] on the [[locally compact Hausdorff spaces]] with [[proper maps]] between them, the [[functor]] of [[one-point compactification]] (Prop. \ref{OnePointCompactificationFunctor})

$$
  (-)^{cpt}
  \;\colon\;
  Top_{LCHaus}
  \longrightarrow
  Top^{\ast/}
$$

sends [[Cartesian products]] ([[product topological spaces]]) to [[smash products]] of [[pointed topological spaces]], hence constitutes a [[strong monoidal functor]], in that there is a [[natural transformation|natural]] [[homeomorphism]]:

$$
  \big(
    X \times Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \wedge Y^{cpt}
  \,.
$$

=--

This is briefly mentioned in, for instance, [Bredon 93, p. 199](#Bredon93).
The argument may be found spelled out in: [MO:a/1645794/](https://math.stackexchange.com/a/1645794/58526), [Cutler 20, Prop. 1.6](#Cutler20).

[[!include smash-monoidal diagonals -- section]]

## Related concepts

* [[pointed simplicial set]]

* [[retractive space]]

* [[loop space]], [[reduced suspension]]

* [[smash product]], [[wedge sum]]

* [[pointed homotopy type]]

* [[spectrum]]

* [[classical model structure on pointed topological spaces]]

## References

Textbook accounts:

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], Chapters IV.4 and V.7 of _[[Calculus of fractions and homotopy theory]]_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967)  ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf))

* {#Bredon93} [[Glen Bredon]], _Topology and Geometry_, Graduate Texts in Mathematics 139, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

Review:

* {#Cutler20} [[Tyrone Cutler]], _The category of pointed topological spaces_ (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory%20II%20-%20The%20Pointed%20Category.pdf), [[CutlerPointedTopologicalSpaces.pdf:file]]]

[[!redirects pointed topological spaces]]

[[!redirects based topological space]]
[[!redirects based topological spaces]]


pointed type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea

A **pointed type** is a [[type]] equipped with a [[term]] of that type.

The [[categorical semantics]] is a [[pointed object]].

## Properties

There is a way of pointing any type $X$ by forming the sum $X+\mathbf{1}$ and taking $inr(\star_{\mathbf{1}})$ as the base point.

### Propositions as types

In the [[propositions as types]] interpretation of [[type theory]], every pointed type represents a [[true]] [[proposition]]. Contrast this to an inhabited type, which only represents the [[double negation]] of a true proposition.

## Type of pointed types

For a given type universe $\mathcal{U}$ the type of pointed types is

$$\mathcal{U}_+\equiv \sum_{X:\mathcal{U}}X$$

## Related entries

* [[maybe monad]]

* [[loop space type]]

* [[reduced suspension type]]

* [[prespectrum type]]

## References

* [[Univalent Foundations Project]], [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

* [[Martín Escardó]], *[Pointed types](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#pointed-types)*, §33.5 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

[[!redirects pointed types]]