5. Limit and Colimit
Many constructions in algebraic topology are described by their universal properties. There are two important ways (dual notions) to define new objects of such types, called the limit and colimit. In this section, we give a brief discussion of them.
Let be a small category (i.e. objects form a set). Let be a category. Recall that we have a functor category (Definition 1.27) where objects are functors from to , and morphisms are natural transformations. We also write
Definition 5.1. We define the diagonal (or constant) functor which assigns to the functor that sends all objects in to and all morphisms to .
Diagram
Let be a diagram, with vertices and arrows. We can define a category still denoted by :
• | vertices in the diagram , |
• | morphisms are composites of all given arrows as well as additional identity maps. |
Example 5.2. The following diagramdefines a category with three objects . There is only one morphism from to , one from to , and one from to which is the composite of the previous two. Given an object , the constant functor can be represented by the following data
Example 5.3. The following diagramdefines a category with three objects . There is only one morphism from to and one from to . There are two morphisms from to , one of them is the composite of the previous two morphisms, and the other one is represented by the arrow .
Example 5.4. The following diagramdefines a category with two objects . Morphisms from to contains the identity , the composition of and and so on.
Given a diagram , a functor is determined by assigning vertices and arrows the corresponding objects and morphisms in . For example, the following datadefine a functor from to . Such a data will be also called a -shaped diagram in .
Limit
Definition 5.5 (Limit). Let be a functor. A limit for is an object in together with a natural transformationsuch that for every object of and every natural transformation , there exists a unique morphism such that . In other words, the following diagram is commutative.
For example, consider the following -shaped diagram in which represents a functor Then its limit is an object that fits into the commutative diagramMoreover for any other object fitting into the same commutative diagram, there exists a unique which makes the following diagram commutative
Proposition 5.6. Let and be two limits of with natural transformations . Then there exists a unique isomorphism in which makes the following diagram commutative
The above proposition says that if the limit of exists, then it is unique up to a canonical isomorphism.
Definition 5.7. We denote the limit of by (if exists).
The universal property of the limit gives the adjunctionThis immediately leads to the following theorem.
Theorem 5.8. Let be a category. Then the following are equivalent
(1) | Every has a limit. |
(2) | The constant functor has a right adjoint. |
In this case, the right adjoint of the constant functor is the limit.
Example 5.9 (Pullback). The limit of the diagram giveswhich is called the pullback.
In the category , the pull-back exists and is given by the subset of
Example 5.10 (Tower and inverse limit). We consider the following category :
• | Objects of are positive integers. |
• | Given , the morphism set is empty if and is a single point if . |
Let be the opposite category of . A functor is represented by the tower diagramThe limit of the tower diagram is also called the inverse limit of the tower and written as :
Theorem 5.11. Letbe adjoint functors. Assume the limit of exists. Then the limit of also exists and is given by In other words, right adjoint functors preserve limits.
Proof. Let . Assume we have a natural transformationBy adjunction, this is equivalent to a natural transformation .
Remark 5.12. A functor is called continuous if it preserves all limits. This theorem says if a functor has a left adjoint, then it is continuous. Under certain conditions, the reverse is also true (Adjoint Functor Theorem).
Corollary 5.13. The forgetful functor preserves limit.
Example 5.14. Consider the following diagram in We would like to understand the pull-back of the above diagram in . By Example 5.9 and Corollary 5.13, we know that the underlying set for (if exists) isIt is not hard to see that if we assign the subspace topology of the topological product , then is indeed the pull-back in . In particular, pull-back exists in .
The next proposition says that fibrations behave well under pull-back. We leave the proof to the reader as exercise.
Proposition 5.15. Let be a fibration, and be continuous. Consider the pull-back diagramThen is also a fibration. In other words, the pull-back of a fibration is a fibration.
Colimit
The notion of colimit is dual to limit.
Definition 5.16 (Colimit). Let . A colimit for is an object in together with a natural transformationsuch that for every object of and every natural transformation , there exists a unique map such that . In other words, the following diagram is commutativeThe colimit, if exists, is unique up to a unique isomorphism, and will be denoted by .
The following theorems are dual to the limit case as well and can be proved dually.
Theorem 5.17. Let be a category. Then the following are equivalent
(1) | Every has a colimit |
(2) | The constant functor has a left adjoint. |
In this case, the left adjoint of the constant functor is the colimits.
Theorem 5.18. Letbe adjoint functors. Assume the colimit of exists. Then the colimit of also exists and is given byIn other words, left adjoint functors preserve colimit.
Remark 5.19. A functor is called co-continuous if it preserves all colimits. This theorem says if a functor has a right adjoint, then it is co-continuous. Under certain conditions, the reverse is also true (Adjoint Functor Theorem).
Corollary 5.20. The forgetful functor preserves colimits.
Example 5.21 (Pushout). The colimit of the diagram givesThis colimit is called the pushout, the dual notion of pullback. It has the following universal property
Here are some examples.
• | Let , in . Their pushout is the quotient of the disjoint union by identifying . It glues along using . For instance, the following picture is an illustration. |
• | Let , be two morphisms in , then their pushout iswhere is the free product and is the normal subgroup generated by . |
Example 5.22 (Telescope and direct limit). A functor is represented by the telescope diagramThe colimit of telescope diagram is also called the direct limit of the telescope and written as :
Product
Definition 5.23. Let be a category, be a set of objects in . Their product is an object in together with satisfying the following universal property: for any in and , there exists a unique morphism such that the following diagram commutes
For product of two objects, we have the following diagram
The product is a limit. In fact, let us equip the index set with the category structure such that it has only identity morphisms. Then the data is the same as a functor . Their product is precisely . In particular, the product is unique up to isomorphism if it exists. We denote it byA useful consequence is that the product is preserved under right adjoint functors (like forgetful functors).
Example 5.24.
• | Let . Then is the Cartesian product. |
• | Let . Then is the Cartesian product with induced product topology. Namely, we have is continuous if and only if are continuous for any . |
• | Let . Then is the Cartesian product with induced group structure, i.e.with . |
Coproduct
Definition 5.25. Let be a category, be a set of objects in . Their coproduct is an object in together with satisfying the following universal property: for any in and , there exists a unique morphism such that the following diagram commutes
The coproduct is a colimit. As in the discussion of product, the data defines a functor . Their coproduct is precisely , which is unique up to isomorphism if it exists. We denote it byA useful consequence is that the coproduct is preserved under left adjoint functors (like free constructions).
Example 5.26.
• | Let . Then is the disjoint union of sets. |
• | Let . Then is the disjoint union of topological spaces. Clearly, continuous maps uniquely extends to . |
• | Let . Then is the free product of groups. More precisely, we havewhereif and is the group production in . The group structure in isGiven group homomorphisms , it uniquely determines the group homomorphismThis is precisely the coproduct property. When there are only finitely many , we will write |
Wedge and smash product
Definition 5.27. We define the category of pointed topological space where
• | an object is a topological space with a based point , |
• | morphisms are based continuous maps that map based point to based point. |
Given a space , we can define a pointed space by adding an extra pointThis defines a functorOn the other hand, we have a forgetful functor by forgetting the base pointThey form an adjoint pairThis implies that the limit in will be the same as the limit in . In particular, the product of pointed spaces in is the topological product
In , the coproduct of two pointed spaces is the wedge product . Specifically,is the quotient of the disjoint union of and by identifying the base points and . The identified based point is the new based point of . In general, we havewhere again identifies all based points in ’s. In other words, is the joining of spaces at a single point.
Example 5.28. The Figure-8 in Example 3.9 can be identified with .
Example 5.29. There is a natural homeomorphismThis implies that . See Figure 1 for case. In this case, the result, i.e. , can be also realized by cutting the green/purple circles on the torus (where we get a square) and gluing them (the boundary of the square) into one point.
Complete and cocomplete
Definition 5.30. A category is called complete (cocomplete) if for any with a small category, the limit (the colimit ) exists.
Example 5.31. are complete and cocomplete.
For example, in , the limit of is given bywhich is a subset of . The colimit is given bywhich is a quotient of .
For another example, we consider . Since the forgetful functor has both a left adjoint and a right adjoint, it preserves both limits and colimits. Given , its limit has the same underlying set as that in above, but equipped with the induced topology from product and subspace. Similarly, the colimit is the quotient of disjoint unions of with the induced quotient topology.
Initial and terminal object
Definition 5.32. An initial/universal object of a category is an object such that for every object in , there exists precisely one morphism . Dually, a terminal/final object satisfies that for every object there exists precisely one morphism . If an object is both initial and terminal, it is called a zero object or null object.
The defining universal property implies that the initial object and the terminal object are unique up to isomorphism if they exist.
Example 5.33. The emptyset is the initial object in , and the set with a single point is the terminal object in . The same is true for .
The limit of a functor can be viewed as a terminal object as follows. We define a category
• | an object of is an object together with a natural transformation |
• | a morphism in is a morphism in such that the following diagram is commutative |
Then is the terminal object in . A dual construction says can be viewed as an initial object.