nLab
cone

Idea

In homotopy theory, the cone of a space $X$ is the space got by taking the $X$-shaped cylinder $X\times I$, where $I$ may be an interval object, and squashing one end down to a point. The eponymous example is where $X$ is the circle, i.e. the topological space $S^1$, and $I$ is the standard interval $[0,1]$. Then the cartesian product $X\times I$ really is a cylinder, and the cone of $X$ is likewise a cone.

This notion also makes sense when $X$ is a category, if $I$ is taken to be the interval category $\{0\to 1\}$, i.e. the ordinal $2$. Note that since the interval category is directed, this gives two different kinds of cone, depending on which end we squash down to a point.

Another, perhaps more common, meaning of ‘cone’ in category theory is that of a cone over (or under) a diagram. This is just a diagram over the cone category, as above. Explicitly, a cone over $F:J\to C$ is an object $c$ in $C$ equipped with a morphism from $c$ to each vertex of $F$, such that every new triangle arising in this way commutes. A cone which is universal is a limit.

In category theory, the word cocone is sometimes used for the case when we squash the other end of the interval; thus $c$ is equipped with a morphism to $c$ from each vertex of $F$ (but $c$ itself still belongs to $C$). A cocone in this sense which is universal is a colimit. However, one should beware that in homotopy theory, the word cocone is used for a different dualization.

This definition generalizes to higher category theory. In particular in (∞,1)-category theory a cone over an ∞-groupoid is essentially a cone in the sense of homotopy theory.

Definition

In homotopy theory

If $X$ is a space, then the cone of $X$ is the homotopy pushout of the identity on $X$ along the unique map to the point:

$$\begin{array}{ccc}X& \to & X\\ \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(X)\end{array}.$$\array{ X & \to & X \\ \downarrow & & \downarrow \\ * & \to & cone(X) }\,.

This homotopy pushout can be computed as the ordinary pushout $\mathrm{cone}(X):=X\times I{⨿}_X*$

$$\begin{array}{ccc}X& \stackrel{d_1}{\to }& X\times I\\ \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(X)\end{array}.$$\array{ X &\stackrel{d_1}{\to} & X \times I \\ \downarrow && \downarrow \\ * &\to& cone(X) } \,.

If $X$ is a simplicial set, then the cone of $X$ is the join of $X$ with the point.

The mapping cone (q.v.) of a morphism $f:X\to Y$ is then the pushout along $f$ of the inclusion $X\to \mathrm{cone}(X)$.

As a monad

In contexts where intervals $I$ can be treated as monoid objects, the cone construction as quotient of a cylinder with one end identified with a point,

$$C(X)=I\times X/(0\times X)\sim p,$$C(X) = I \times X/(0 \times X) \sim p,

carries a structure of monad $C$. In such cases, the monoid has a multiplicative identity $1$ and an absorbing element $0$, where multiplication by $0$ is the constant map at $0$. In that case, a $C$-algebra consists of an object $X$ together with

• An action of the monoid, $a:I\times X\to X$.

• A constant or basepoint $x_0:1\to X$

such that $a(0,x)=x_0$ for all $x$. This equation can be expressed in any category $C$ with finite products and a suitable interval object $I$ as monoid (for example, $\mathrm{Top}$, where $I=[0,1]$ is a monoid under real multiplication, or under $\min$ as multiplication). Under some reasonable assumptions (e.g., if the $C$ has quotients, and these are preserved by the functor $I\times -$), the category of $C$-algebras will be monadic over $C$ and the free $C$-algebra on $X$ will be $C(X)$ as described above. The category of $C$-algebras will also be monadic over the category of pointed $C$-objects, $1\downarrow C$.

These observations apply for example to $\mathrm{Top}$, and also to $\mathrm{Cat}$ where the interval category $2$ is a monoid in $\mathrm{Cat}$ under the $\min$ operation (see below).

If in addition the underlying category $C$ is cartesian closed, or more generally if $I$ is exponentiable, the monad $C$ on pointed $C$-objects also has a right adjoint $P$ which can be regarded as a path space construction $P$, where we have a pullback

$$\begin{array}{ccc}P(X)& \to & 1\\ \downarrow & & \downarrow \\ X^I& \stackrel{{\mathrm{eval}}_0}{\to }& X.\end{array}$$\array{ P(X) & \to & 1 \\ \downarrow & & \downarrow \\ X^I & \stackrel{eval_0}{\to} & X. }

For general abstract reasons, the right adjoint $P$ carries a comonad structure whereby $C$-algebras are equivalent to $P$-coalgebras. Considered over the category of simplicial sets, this is closely connected to decalage.

In category theory

If $C$ is a category, then the cone of $C$ is the cocomma category? of the identity on $C$ and the unique map to the terminal category:

$$\begin{array}{ccc}C& \to & C\\ \downarrow & \Rightarrow & \downarrow \\ *& \to & \mathrm{cone}(C)\end{array}.$$\array{ C & \to & C \\ \downarrow & \Rightarrow & \downarrow \\ * & \to & cone(C) }\,.

Again, this may be computed as a pushout:

$$\begin{array}{ccc}C& \stackrel{d_1}{\to }& C\times 2\\ \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(C)\end{array}.$$\array{ C &\stackrel{d_1}{\to} & C \times \mathbf{2} \\ \downarrow && \downarrow \\ * &\to& cone(C) } \,.

The cone of $C$ may equivalently be thought of, or defined, as the result of adjoining a new initial object to $C$.

Cones over a diagram

A cone in a category $C$ is given by a category $J$ together with a functor $\mathrm{cone}(J)\to C$. By the universal property of the cocomma category, to give such a functor is to give an object $c$ of $C$, a functor $F:J\to C$, and a natural transformation

$$T:\Delta (c)\to F$$T: \Delta(c) \to F

where $\Delta (c):J\to C$ denotes the constant functor at the object $c$. Such a transformation is called a cone over the diagram $F$.

In other words, a cone consists of morphisms (called the components of the cone)

$$T_j:c\to F(j),$$T_j: c \to F(j),

one for each object $j$ of $J$, which are compatible with all the morphisms $F(f):F(j)\to F(k)$ of the diagram, in the sense that each diagram

$$\begin{array}{ccccc}& & c& & \\ & T_j\swarrow & & \searrow T_k& \\ F(j)& & \stackrel{F(f)}{⟶}& & F(k)\\ \end{array}$$\array{ {}&{}&c&{}&{} \\ {}& \mathllap{\scriptsize{T_j}}\swarrow &{}& \searrow\mathrlap{\scriptsize{T_k}} &{} \\ F(j) &{}&\stackrel{F(f)}{\longrightarrow} &{}& F(k) \\ }

commutes.

It’s called a cone because one pictures $c$ as sitting at the vertex, and the diagram itself as forming the base of the cone.

A cocone in $C$ is precisely a cone in the opposite category $C^{\mathrm{op}}$.

Over a diagram in an $(\mathrm{\infty },1)$-category

For $F:D\to C$ a diagram of (∞,1)-categories, i.e. an (∞,1)-functor, the $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },1)$-cones over $F$ is the over quasi-category denoted $C_{/F}$. Its objects are cones over $F$. Its k-morphisms are $k$-homotopies between cones. The (∞,1)-categorical limit over $F$ is, if it exists, the initial object in $C_{/F}$.

See also

These are shaped like the homotopy-theoretic cone, so maybe there is a deeper relationship:

• positive cone (in an ordered group, such as an operator algebra),
• future cone (of an event? in a Lorentzian manifold, such as spacetime),
• convex cone (in a vector space).