The Stacks project

4.14 Limits and colimits

Let $\mathcal{C}$ be a category. A diagram in $\mathcal{C}$ is simply a functor $M : \mathcal{I} \to \mathcal{C}$. We say that $\mathcal{I}$ is the index category or that $M$ is an $\mathcal{I}$-diagram. We will use the notation $M_ i$ to denote the image of the object $i$ of $\mathcal{I}$. Hence for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $M(\phi ) : M_ i \to M_{i'}$.

Definition 4.14.1. A limit of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ in $\mathcal{C}$ together with morphisms $p_ i : \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \to M_ i$ such that

  1. for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $p_{i'} = M(\phi ) \circ p_ i$, and

  2. for any object $W$ in $\mathcal{C}$ and any family of morphisms $q_ i : W \to M_ i$ (indexed by $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $q_{i'} = M(\phi ) \circ q_ i$ there exists a unique morphism $q : W \to \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ such that $q_ i = p_ i \circ q$ for every object $i$ of $\mathcal{I}$.

Limits $(\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M, (p_ i)_{i\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})})$ are (if they exist) unique up to unique isomorphism by the uniqueness requirement in the definition. Products of pairs, fibre products, and equalizers are examples of limits. The limit over the empty diagram is a final object of $\mathcal{C}$. In the category of sets all limits exist. The dual notion is that of colimits.

Definition 4.14.2. A colimit of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M$ in $\mathcal{C}$ together with morphisms $s_ i : M_ i \to \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M$ such that

  1. for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $s_ i = s_{i'} \circ M(\phi )$, and

  2. for any object $W$ in $\mathcal{C}$ and any family of morphisms $t_ i : M_ i \to W$ (indexed by $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $t_ i = t_{i'} \circ M(\phi )$ there exists a unique morphism $t : \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \to W$ such that $t_ i = t \circ s_ i$ for every object $i$ of $\mathcal{I}$.

Colimits $(\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M, (s_ i)_{i\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})})$ are (if they exist) unique up to unique isomorphism by the uniqueness requirement in the definition. Coproducts of pairs, pushouts, and coequalizers are examples of colimits. The colimit over an empty diagram is an initial object of $\mathcal{C}$. In the category of sets all colimits exist.

Remark 4.14.3. The index category of a (co)limit will never be allowed to have a proper class of objects. In this project it means that it cannot be one of the categories listed in Remark 4.2.2

Remark 4.14.4. We often write $\mathop{\mathrm{lim}}\nolimits _ i M_ i$, $\mathop{\mathrm{colim}}\nolimits _ i M_ i$, $\mathop{\mathrm{lim}}\nolimits _{i\in \mathcal{I}} M_ i$, or $\mathop{\mathrm{colim}}\nolimits _{i\in \mathcal{I}} M_ i$ instead of the versions indexed by $\mathcal{I}$. Using this notation, and using the description of limits and colimits of sets in Section 4.15 below, we can say the following. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram.

  1. The object $\mathop{\mathrm{lim}}\nolimits _ i M_ i$ if it exists satisfies the following property

    \[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, \mathop{\mathrm{lim}}\nolimits _ i M_ i) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \]

    where the limit on the right takes place in the category of sets.

  2. The object $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ if it exists satisfies the following property

    \[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(\mathop{\mathrm{colim}}\nolimits _ i M_ i, W) = \mathop{\mathrm{lim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \]

    where on the right we have the limit over the opposite category with value in the category of sets.

By the Yoneda lemma (and its dual) this formula completely determines the limit, respectively the colimit.

Remark 4.14.5. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. In this setting a cone for $M$ is given by an object $W$ and a family of morphisms $q_ i : W \to M_ i$, $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ such that for all morphisms $\phi : i \to i'$ of $\mathcal{I}$ the diagram

\[ \xymatrix{ & W \ar[dl]_{q_ i} \ar[dr]^{q_{i'}} \\ M_ i \ar[rr]^{M(\phi )} & & M_{i'} } \]

is commutative. The collection of cones forms a category with an obvious notion of morphisms. Clearly, the limit of $M$, if it exists, is a final object in the category of cones. Dually, a cocone for $M$ is given by an object $W$ and a family of morphisms $t_ i : M_ i \to W$ such that for all morphisms $\phi : i \to i'$ in $\mathcal{I}$ the diagram

\[ \xymatrix{ M_ i \ar[rr]^{M(\phi )} \ar[dr]_{t_ i} & & M_{i'} \ar[dl]^{t_{i'}} \\ & W } \]

commutes. The collection of cocones forms a category with an obvious notion of morphisms. Similarly to the above the colimit of $M$ exists if and only if the category of cocones has an initial object.

As an application of the notions of limits and colimits we define products and coproducts.

Definition 4.14.6. Suppose that $I$ is a set, and suppose given for every $i \in I$ an object $M_ i$ of the category $\mathcal{C}$. A product $\prod _{i\in I} M_ i$ is by definition $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ (if it exists) where $\mathcal{I}$ is the category having only identities as morphisms and having the elements of $I$ as objects.

An important special case is where $I = \emptyset $ in which case the product is a final object of the category. The morphisms $p_ i : \prod M_ i \to M_ i$ are called the projection morphisms.

Definition 4.14.7. Suppose that $I$ is a set, and suppose given for every $i \in I$ an object $M_ i$ of the category $\mathcal{C}$. A coproduct $\coprod _{i\in I} M_ i$ is by definition $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M$ (if it exists) where $\mathcal{I}$ is the category having only identities as morphisms and having the elements of $I$ as objects.

An important special case is where $I = \emptyset $ in which case the coproduct is an initial object of the category. Note that the coproduct comes equipped with morphisms $M_ i \to \coprod M_ i$. These are sometimes called the coprojections.

Lemma 4.14.8. Suppose that $M : \mathcal{I} \to \mathcal{C}$, and $N : \mathcal{J} \to \mathcal{C}$ are diagrams whose colimits exist. Suppose $H : \mathcal{I} \to \mathcal{J}$ is a functor, and suppose $t : M \to N \circ H$ is a transformation of functors. Then there is a unique morphism

\[ \theta : \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \longrightarrow \mathop{\mathrm{colim}}\nolimits _\mathcal {J} N \]

such that all the diagrams

\[ \xymatrix{ M_ i \ar[d]_{t_ i} \ar[r] & \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \ar[d]^{\theta } \\ N_{H(i)} \ar[r] & \mathop{\mathrm{colim}}\nolimits _\mathcal {J} N } \]

commute.

Proof. Omitted. $\square$

Lemma 4.14.9. Suppose that $M : \mathcal{I} \to \mathcal{C}$, and $N : \mathcal{J} \to \mathcal{C}$ are diagrams whose limits exist. Suppose $H : \mathcal{I} \to \mathcal{J}$ is a functor, and suppose $t : N \circ H \to M$ is a transformation of functors. Then there is a unique morphism

\[ \theta : \mathop{\mathrm{lim}}\nolimits _\mathcal {J} N \longrightarrow \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \]

such that all the diagrams

\[ \xymatrix{ \mathop{\mathrm{lim}}\nolimits _\mathcal {J} N \ar[d]^{\theta } \ar[r] & N_{H(i)} \ar[d]_{t_ i} \\ \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \ar[r] & M_ i } \]

commute.

Proof. Omitted. $\square$

Lemma 4.14.10. Let $\mathcal{I}$, $\mathcal{J}$ be index categories. Let $M : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$ be a functor. Assume that $M_{i, \infty } = \mathop{\mathrm{colim}}\nolimits _ j M_{i,j}$ exists for all $i$. Then the resulting functor $M_{-, \infty } : \mathcal{I} \to \mathcal{C}$ has a colimit if and only if $M$ does, and then the colimits coincide. In particular, we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ j M_{i, j} = \mathop{\mathrm{colim}}\nolimits _{i, j} M_{i, j} = \mathop{\mathrm{colim}}\nolimits _ j \mathop{\mathrm{colim}}\nolimits _ i M_{i, j} \]

provided all the indicated colimits exist. Similar for limits.

Proof. Omitted. $\square$

Lemma 4.14.11. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. Write $I = \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and $A = \text{Arrows}(\mathcal{I})$. Denote $s, t : A \to I$ the source and target maps. Suppose that $\prod _{i \in I} M_ i$ and $\prod _{a \in A} M_{t(a)}$ exist. Suppose that the equalizer of

\[ \xymatrix{ \prod _{i \in I} M_ i \ar@<1ex>[r]^\phi \ar@<-1ex>[r]_\psi & \prod _{a \in A} M_{t(a)} } \]

exists, where the morphisms are determined by their components as follows: $p_ a \circ \psi = M(a) \circ p_{s(a)}$ and $p_ a \circ \phi = p_{t(a)}$. Then this equalizer is the limit of the diagram.

Proof. Omitted. $\square$

slogan

Lemma 4.14.12. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. Write $I = \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and $A = \text{Arrows}(\mathcal{I})$. Denote $s, t : A \to I$ the source and target maps. Suppose that $\coprod _{i \in I} M_ i$ and $\coprod _{a \in A} M_{s(a)}$ exist. Suppose that the coequalizer of

\[ \xymatrix{ \coprod _{a \in A} M_{s(a)} \ar@<1ex>[r]^\phi \ar@<-1ex>[r]_\psi & \coprod _{i \in I} M_ i } \]

exists, where the morphisms are determined by their components as follows: The component $M_{s(a)}$ maps via $\psi $ to the component $M_{t(a)}$ via the morphism $M(a)$. The component $M_{s(a)}$ maps via $\phi $ to the component $M_{s(a)}$ by the identity morphism. Then this coequalizer is the colimit of the diagram.

Proof. Omitted. $\square$

Comments (7)

Comment #3181 by Diego Antonio on

I have a question. Other references define limit by means of a contravariant functor. By the way limits and colimits are defined in this tag, one can take/form the limit and colimit of the same functor. Is this correct? I mean, doing this will imply that there's always a map from the limit to the colimit. Then whenever the category has an initial and a terminal object, there will always be a map from the terminal object to the initial one. This map would then be an isomorphism, which is not always the case as in Set.

Maybe I'm missing something.

Comment #3183 by on

What you say is incorrect. Suggest computing some examples.

Comment #3195 by DiegoAntonio on

Yes! I realized what was wrong. I was very confused at the moment.

Comment #7083 by Samuel Tiersma on

Statement of Lemma 002P, definition of : the morphism should be .

Comment #8885 by jhzg on

How to prove Lemma 002M? Is this lemma certain to be correct？

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