cone in nLab

Contents

This entry is mostly about cones in homotopy theory and category theory. For more geometric cones see at cone (Riemannian geometry).

Contents

Idea

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

This notion also makes sense when XX is a category, if II is taken to be the interval category {0→1}\{ 0 \to 1 \}, i.e. the ordinal 2\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:J→CF\colon J \to C is an object cc in CC equipped with a morphism from cc to each vertex of FF, 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 cc is equipped with a morphism to cc from each vertex of FF (but cc itself still belongs to CC). 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 XX is a space, then the cone of XX is the homotopy pushout of the identity on XX along the unique map to the point:

X → X ↓ ↓ * → cone(X). \array{ X & \to & X \\ \downarrow & & \downarrow \\ * & \to & cone(X) }\,.

This homotopy pushout can be computed as the ordinary pushout cone(X):=X×I⨿ X*cone(X) := X\times I \amalg_X *

X →d 1 X×I ↓ ↓ * → cone(X). \array{ X &\stackrel{d_1}{\to} & X \times I \\ \downarrow && \downarrow \\ * &\to& cone(X) } \,.

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

The mapping cone (q.v.) of a morphism f:X→Yf \colon X \to Y is then the pushout along ff of the inclusion X→cone(X)X \to cone(X).

As a monad

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

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

carries a structure of monad CC. In such cases, the monoid has a multiplicative identity 11 and an absorbing element 00, where multiplication by 00 is the constant map at 00. In that case, a CC-algebra consists of an object XX together with

• An action of the monoid, a:I×X→Xa: I \times X \to X.

• A constant or basepoint x 0:1→Xx_0 \colon 1 \to X

such that a(0,x)=x 0a(0, x) = x_0 for all xx. This equation can be expressed in any category C\mathbf{C} with finite products and a suitable interval object II as monoid (for example, TopTop, where I=[0,1]I = [0, 1] is a monoid under real multiplication, or under minmin as multiplication). Under some reasonable assumptions (e.g., if the C\mathbf{C} has quotients, and these are preserved by the functor I×−I \times -), the category of CC-algebras will be monadic over C\mathbf{C} and the free CC-algebra on XX will be C(X)C(X) as described above. The category of CC-algebras will also be monadic over the category of pointed C\mathbf{C}-objects, 1↓C1 \downarrow \mathbf{C}.

These observations apply for example to TopTop, and also to CatCat where the interval category 2\mathbf{2} is a monoid in CatCat under the minmin operation (see below).

If in addition the underlying category C\mathbf{C} is cartesian closed, or more generally if II is exponentiable, the monad CC on pointed C\mathbf{C}-objects also has a right adjoint PP which can be regarded as a path space construction PP, where we have a pullback

P(X) → 1 ↓ ↓ X I →eval 0 X.\array{ P(X) & \to & 1 \\ \downarrow & & \downarrow \\ X^I & \stackrel{eval_0}{\to} & X. }

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

In category theory

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

C → C ↓ ⇒ ↓ * → cone(C). \array{ C & \to & C \\ \downarrow & \Rightarrow & \downarrow \\ * & \to & cone(C) }\,.

Again, this may be computed as a pushout:

C →d 1 C×2 ↓ ↓ * → cone(C). \array{ C &\stackrel{d_1}{\to} & C \times \mathbf{2} \\ \downarrow && \downarrow \\ * &\to& cone(C) } \,.

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

Cones over a diagram

A cone in a category CC is given by a category JJ together with a functor cone(J)→Ccone(J) \to C. By the universal property of the cocomma category, to give such a functor is to give an object cc of CC, a functor F:J→CF \colon J \to C, and a natural transformation

T:Δ(c)→FT: \Delta(c) \to F

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

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

T j:c⟶F(j), T_j \;\colon\; c \longrightarrow F(j) \,,

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

c T j↙ ↘T k F(j) ⟶F(f) F(k) \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 cc as sitting at the vertex, and the diagram itself as forming the base of the cone.

A morphism of such cones is a natural transformation α:Δ(c)→Δ(c′)\alpha\colon \Delta(c)\to\Delta(c') such that the diagram

Δ(c) ⟶α Δ(c′) T↘ ↙T′ F \array{ \Delta(c) &{}&\overset{\alpha}{\longrightarrow} &{}& \Delta(c') \\ {}& \mathllap{\scriptsize{T}}\searrow &{}& \swarrow\mathrlap{\scriptsize{T'}} &{} \\ {}&{}&F&{}&{} }

commutes. Note that naturality of any such α\alpha implies that for all i,j∈Ji,j\in J, α i=α j\alpha_i=\alpha_j, so that α=Δ(ϕ)\alpha=\Delta(\phi) for some ϕ:c→c′\phi \colon c \to c' in CC. The single component ϕ\phi itself is often referred to as the cone morphism.

An equivalent definition of a cone morphism ϕ:T→T′\phi : T \to T' says that all component diagrams

c ⟶ϕ c′ T j↘ ↙T′ j F(j) \array{ c &{}& \overset{\phi}{\longrightarrow} &{}& c' \\ {}& \mathllap{\scriptsize{T_j}}\searrow &{}& \swarrow\mathrlap{\scriptsize{T'_j}} &{} \\ {}&{}&F(j)&{}&{} }

commute.

Cones and their morhisms over a given diagram JJ clearly form a category. The terminal object in this category, if it exists, is the limit of the diagram (see there).

A cocone in CC is precisely a cone in the opposite category C opC^{op}.

Over a diagram in an (∞,1)(\infty,1)-category

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

See also

• cone type

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).