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 $\mathbf{2}$. 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 $F\colon J \to C$ 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.
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:
This homotopy pushout can be computed as the ordinary pushout $cone(X) := X\times I \amalg_X *$
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 $f \colon X \to Y$ is then the pushout along $f$ of the inclusion $X \to cone(X)$.
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,
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 $x_0 \colon 1 \to X$
such that $a(0, x) = x_0$ for all $x$. This equation can be expressed in any category $\mathbf{C}$ with finite products and a suitable interval object $I$ as monoid (for example, $Top$, where $I = [0, 1]$ is a monoid under real multiplication, or under $min$ as multiplication). Under some reasonable assumptions (e.g., if the $\mathbf{C}$ has quotients, and these are preserved by the functor $I \times -$), the category of $C$-algebras will be monadic over $\mathbf{C}$ 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 $\mathbf{C}$-objects, $1 \downarrow \mathbf{C}$.
These observations apply for example to $Top$, and also to $Cat$ where the interval category $\mathbf{2}$ is a monoid in $Cat$ under the $min$ operation (see below).
If in addition the underlying category $\mathbf{C}$ is cartesian closed, or more generally if $I$ is exponentiable, the monad $C$ on pointed $\mathbf{C}$-objects also has a right adjoint $P$ which can be regarded as a path space construction $P$, where we have a pullback
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.
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:
Again, this may be computed as a pushout:
The cone of $C$ may equivalently be thought of, or defined, as the result of adjoining a new initial object to $C$.
A cone in a category $C$ is given by a category $J$ together with a functor $cone(J) \to C$. By the universal property of the cocomma category, to give such a functor is to give an object $c$ of $C$, a functor $F \colon J \to C$, and a natural transformation
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)
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
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 $C^{op}$.
For $F : D \to C$ a diagram of (∞,1)-categories, i.e. an (∞,1)-functor, the $(\infty,1)$-category of $(\infty,1)$-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 $C_{/F}$.
These are shaped like the homotopy-theoretic cone, so maybe there is a deeper relationship: