In homotopy theory, the cone of a space is the space got by taking the -shaped cylinder , where may be an interval object, and squashing one end down to a point. The eponymous example is where is the circle, i.e. the topological space , and is the standard interval . Then the cartesian product really is a cylinder, and the cone of is likewise a cone.
This notion also makes sense when is a category, if is taken to be the interval category , i.e. the ordinal . 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 is an object in equipped with a morphism from to each vertex of , 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 is equipped with a morphism to from each vertex of (but itself still belongs to ). 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.
If is a space, then the cone of is the homotopy pushout of the identity on along the unique map to the point:
This homotopy pushout can be computed as the ordinary pushout
The mapping cone (q.v.) of a morphism is then the pushout along of the inclusion .
carries a structure of monad . In such cases, the monoid has a multiplicative identity and an absorbing element , where multiplication by is the constant map at . In that case, a -algebra consists of an object together with
An action of the monoid, .
A constant or basepoint
such that for all . This equation can be expressed in any category with finite products and a suitable interval object as monoid (for example, , where is a monoid under real multiplication, or under as multiplication). Under some reasonable assumptions (e.g., if the has quotients, and these are preserved by the functor ), the category of -algebras will be monadic over and the free -algebra on will be as described above. The category of -algebras will also be monadic over the category of pointed -objects, .
These observations apply for example to , and also to where the interval category is a monoid in under the operation (see below).
If in addition the underlying category is cartesian closed, or more generally if is exponentiable, the monad on pointed -objects also has a right adjoint which can be regarded as a path space construction , where we have a pullback
For general abstract reasons, the right adjoint carries a comonad structure whereby -algebras are equivalent to -coalgebras. Considered over the category of simplicial sets, this is closely connected to decalage.
If is a category, then the cone of is the cocomma category? of the identity on and the unique map to the terminal category:
Again, this may be computed as a pushout:
The cone of may equivalently be thought of, or defined, as the result of adjoining a new initial object to .
A cone in a category is given by a category together with a functor . By the universal property of the cocomma category, to give such a functor is to give an object of , a functor , and a natural transformation
where denotes the constant functor at the object . Such a transformation is called a cone over the diagram .
In other words, a cone consists of morphisms (called the components of the cone)
one for each object of , which are compatible with all the morphisms of the diagram, in the sense that each diagram
It’s called a cone because one pictures as sitting at the vertex, and the diagram itself as forming the base of the cone.
For a diagram of (∞,1)-categories, i.e. an (∞,1)-functor, the -category of -cones over is the over quasi-category denoted . Its objects are cones over . Its k-morphisms are -homotopies between cones. The (∞,1)-categorical limit over is, if it exists, the initial object in .
These are shaped like the homotopy-theoretic cone, so maybe there is a deeper relationship: