Lemma. colimits commute with colimits [leinster2016basic, 6.2.10] [tt-0049]
✍️source

Dual to Lemma [tt-0047], colimits commute with colimits.

References

Reference. Basic category theory [leinster2016basic]

2016
Tom Leinster
https://arxiv.org/abs/1612.09375

@article{leinster2016basic,
 title = {Basic category theory},
 author = {Leinster, Tom},
 year = {2016},
 url = {https://arxiv.org/abs/1612.09375},
 journal = {arXiv preprint arXiv:1612.09375}
}

Context

Interactions [tt-002P]

Remark. interactions [leinster2016basic] [tt-0043]

In this section, we will discuss the interactions between

(co)limits
adjunctions
representables

and their relationships with universal properties.

Lemma. adjunction preserves (co)limits [leinster2016basic, 6.3.1] [tt-002O]

Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), \(\mathscr {L}\) preserves colimits, and \(\mathscr {R}\) preserves limits.

Explictly, given \(\mathscr {D}: {\cal J} \to {\cal C}\), we have \[(\operatorname {colim} \mathscr {D}) \mathbin {\bullet } \mathscr {L} \cong \operatorname {colim}(\mathscr {D} \mathbin {\bullet } \mathscr {L})\] and given \(\mathscr {D}': {\cal J} \to {\cal D}\), we have \[(\lim \mathscr {D}') \mathbin {\bullet } \mathscr {R} \cong \lim (\mathscr {D}' \mathbin {\bullet } \mathscr {R})\]

Definition [tt-002H]

Corollary. a representation is a universal element [leinster2016basic, 4.3.2] [tt-0035]

Let \({\cal C}\) be a locally small category and \(\mathscr {F} : {\cal C}^{op} \to \mathbf {Set}\). Then a representation of \(\mathscr {F}\) consists of a pair \((X, u)\) such that the diagram

$\text{[math]}$ commutes.

Remark. universal element [leinster2016basic, 4.3.2] [tt-0036]

Pairs \((Y, y)\) with \(Y \in {\cal C}\) and \(y \in \mathscr {F}(Y)\) in Corollary [tt-0035] are sometimes called elements of the presheaf \(\mathscr {F}\).

Indeed, Lemma [tt-002N] (Yoneda) tells us that \(y\) amounts to a generalized element of \(\mathscr {F}\) of shape \(\mathscr {H}_Y\).

An element \(u\) satisfying condition in Corollary [tt-0035] is sometimes called a universal element of \(\mathscr {F}\). So, Corollary [tt-0035] says that a representation of a presheaf \(\mathscr {F}\) amounts to a universal element of \(\mathscr {F}\).

Lemma. adjunction and representable [leinster2016basic, 4.1.11] [tt-003I]

Any set-valued functor with a left adjoint is representable.

Definition. cone as a natural transformation [leinster2016basic, eq. 6.1] [tt-0040]

Now, given a diagram \(\mathscr {D}: {\cal J} \to {\cal C}\) and an object \(V \in {\cal C}\), a cone on \(\mathscr {D}\) with vertex \(V\) is simply a natural transformation from the diagonal functor \(\Delta _V\) to the diagram \(\mathscr {D}\).

$\text{[math]}$

Writing \(\operatorname {Cone}(V, \mathscr {D})\) for the set of cones on \(\mathscr {D}\) with vertex \(V\), we therefore have \[ \operatorname {Cone}(V, \mathscr {D})=[{\cal J}, {\cal C}] (\Delta _V, \mathscr {D}) . \]

Thus, \(\operatorname {Cone}(V, \mathscr {D})\) is functorial in \(V\) (contravariantly) and \(\mathscr {D}\) (covariantly).

Lemma. limit via representation [leinster2016basic, 6.1.1] [tt-003Y]

Let \({\cal J}\) be a small category, \({\cal C}\) a category, and \(\mathscr {D}: {\cal J} \to {\cal C}\) a diagram. Then there is a one-to-one correspondence between

limit cones on \(\mathscr {D}\)
representations of the natural transformation Cone

with the representing objects being the limit objects (i.e. the vertices) of \(\mathscr {D}\).

Briefly put: a limit \((V, \pi )\) of \(\mathscr {D}\) is a representation of \([{\cal J}, {\cal C}] (\Delta _{-}, \mathscr {D})\).

Diagramatically,

$\text{[math]}$

It implies that \[\operatorname {Cone}(\mathrm {-}, \mathscr {D}) \cong {\cal C}\left (\mathrm {-}, \lim \limits _{\leftarrow {\cal J}} \mathscr {D} \right )\] for any \(\mathrm {-} \in {\cal C}\).

Lemma. representables preserve limits [leinster2016basic, 6.2.2] [tt-0042]

Let \(\mathscr {A}\) be a locally small category and \(X \in {\cal C}\). Then \({\cal C}(X,-): {\cal C} \to \mathbf {Set}\) preserves limits.

Proof.

It follows from Lemma [tt-003Y] and that [leinster2016basic, 6.2.1] \[\operatorname {Cone}(X, \mathscr {D}) \cong \lim \limits _{\leftarrow {\cal J}} {\cal C}(X, \mathscr {D})\] naturally in \(X\) and \(\mathscr {D}\).

Lemma. limits commute with limits [leinster2016basic, 6.2.8] [tt-0047]

Let \({\cal I}\) and \({\cal J}\) be small categories. Let \({\cal C}\) be a locally small category with limits of shape \({\cal I}\) and of shape \({\cal J}\).

Define \[\begin {array}{llll} \mathscr {D}^{\bullet }: & {\cal I} & \to & {[{\cal J}, {\cal C}]} \\ & I & \mapsto & \mathscr {D}(I,-) \end {array}\] and \[\begin {array}{rrrr} \mathscr {D}_{\bullet }: & {\cal J} & \to & {[{\cal I}, {\cal C}]} \\ & J & \mapsto & \mathscr {D}(-, J) \end {array}\]

Then for all \(\mathscr {D}: {\cal I} \times {\cal J} \to {\cal C}\), we have \[ \lim _{\leftarrow {\cal J}} \lim _{\leftarrow {\cal I}} \mathscr {D}^{\bullet } \cong \lim _{\leftarrow {\cal I} {\cal J}} \mathscr {D} \cong \lim _{\leftarrow {\cal I} \leftarrow {\cal J}} \mathscr {D}_{\bullet } \] and all these limits exist. In particular, \({\cal C}\) has limits of shape \({\cal I} \times {\cal J}\).

Lemma. colimits commute with colimits [leinster2016basic, 6.2.10] [tt-0049]
✍️source

Dual to Lemma [tt-0047], colimits commute with colimits.

Remark. [leinster2016basic, 6.2.10] [tt-004A]

Limits do not in general commute with colimits.

Some special cases where they do:

filtered colimits commute with finite limits [stacks2017stacks, 002W].

Lemma. initial and terminal objects via adjunction [leinster2016basic, 2.1.9] [tt-0054]

Initial and terminal objects can be described as adjoints. Let \({\cal C}\) be a category. There exist the unique functor \(! : {\cal C} \to \mathbf {1}\), and a constant object functor \(X : 1 \to {\cal C}\) for each object \(X\).

A left adjoint to \(!\) is exactly an initial object of \({\cal C}\): \[ \mathrm {0} \dashv \ ! : \mathbf {1} \rightleftarrows {\cal C} \]

Similarly, a right adjoint to \(!\) is exactly a terminal object of \({\cal C}\): \[ ! \dashv \mathrm {1} : {\cal C} \rightleftarrows \mathbf {1} \]

Proof.

In both cases, being an adjunction gives an isomorphism for each object \(X\), one side of the isomorphism is \(\mathbf {1}(\mathrm {*}, \mathrm {*})\) which is just \(\mathit {1}_{\mathrm {*}}\), and the other side are \({\cal C}(\mathrm {0}, X)\) or \({\cal C}(X, \mathrm {1})\), and the isomorphism establishes the uniqueness of the arrows (from \(\mathrm {0}\) or to \(\mathrm {1}\)) for each object. The initial or terminal object exists if the corresponding adjunction exists.

Definition [tt-000I]Definition [tt-001Q]Definition [tt-003Z]Definition [tt-003H]

Lemma. (co)limits via adjunction [rosiak2022sheaf, example 200] [tt-005A]

(Co)limits can be phrased entirely in terms of adjunctions:

$\text{[math]}$

The advantages of this adjunction perspective is that the (co)limit of every \({\cal J}\)-shaped diagram in \({\cal C}\) can be defined all at once.

Related

Lemma. limits commute with limits [leinster2016basic, 6.2.8] [tt-0047]

Let \({\cal I}\) and \({\cal J}\) be small categories. Let \({\cal C}\) be a locally small category with limits of shape \({\cal I}\) and of shape \({\cal J}\).

Define \[\begin {array}{llll} \mathscr {D}^{\bullet }: & {\cal I} & \to & {[{\cal J}, {\cal C}]} \\ & I & \mapsto & \mathscr {D}(I,-) \end {array}\] and \[\begin {array}{rrrr} \mathscr {D}_{\bullet }: & {\cal J} & \to & {[{\cal I}, {\cal C}]} \\ & J & \mapsto & \mathscr {D}(-, J) \end {array}\]

Then for all \(\mathscr {D}: {\cal I} \times {\cal J} \to {\cal C}\), we have \[ \lim _{\leftarrow {\cal J}} \lim _{\leftarrow {\cal I}} \mathscr {D}^{\bullet } \cong \lim _{\leftarrow {\cal I} {\cal J}} \mathscr {D} \cong \lim _{\leftarrow {\cal I} \leftarrow {\cal J}} \mathscr {D}_{\bullet } \] and all these limits exist. In particular, \({\cal C}\) has limits of shape \({\cal I} \times {\cal J}\).