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 $X\to \mathrm{cone}(X)$.

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 $C_{/F}$.