Category theory

Limits and colimits



In homotopy theory, the cone of a space XX is the space got 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 {01}\{ 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:JCF\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.


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:XYf \colon X \to Y is then the pushout along ff of the inclusion Xcone(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×XXa: I \times X \to X.

  • A constant or basepoint x 0:1Xx_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, 1C1 \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:JCF \colon J \to C, and a natural transformation

T:Δ(c)FT: \Delta(c) \to F

where Δ(c):JC\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:cF(j),T_j: c \to 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) \\ }


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 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:DCF : 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 initial object in C /FC_{/F}.

See also

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

Revised on April 26, 2013 20:54:44 by Todd Trimble (