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Limits

Modern POV: realize in terms of a Kan extension:

Separating limits vs colimits:

| Limit | Colimit | |

–––––––– | –––––––– | | Inverse | Direct | | \(\lim\) | \(\colim\) | | terminal cones | initial cocones | | terminal | initial | | equalizer | coequalizer | | kernel | cokernel | | pullback | pushout | | product \(\prod\) | coproduct \(\displaystyle\coprod\), direct sum \(\bigoplus\) | | \(\hom({-}, ?)\) | \(\hom(?, {-})\) | | continuous | cocontinuous | | | |

Commuting Limits

• Limits commmute with limits
• Colimits commute with colimits
• Limits need not commute with colimits in general.
• Filtered colimits commute with finite limits in \({\mathsf{Set}}\), and any category with a functor \(\mathsf{C}\to {\mathsf{Set}}\) which is faithful and preserves and reflects finite limits and filtered colimits
• Finite colims of compact are compact
• RAPL: right adjoints preserve limits, and left adjoints preserve colimits.
• Contravariant hom preserves lims: \([A, \lim B_i] = \lim [A, B_i]\).
• Covariant hom preserves colims: \([\lim A_i, B]\cong \colim [A, B_i]\).
• There is a morphism \(\colim \lim A_{i, j} \to \lim \colim A_{i, j}\).
• Limits respect finite group actions, i.e. \((\lim A_i)/G { \, \xrightarrow{\sim}\, }\lim (A_i/G)\).
• Left adjoints are right-exact (LARE), right adjoints are left-exact (RALE)

Definitions

Limits

Colimits

Slogan:build new objects by “gluing together” existing ones.

Definition in terms of cones and cocones:

For \(F: \mathsf{I} \to \mathsf{C}\), define \(\chi_X: \mathsf{I}\to \mathsf{C}\) for \(X\in \mathsf{C}\) regarded as a set (groupoid with only identity morphisms) to be the constant functor \(Y\mapsto X\) for all \(Y\) and \(f\mapsto \operatorname{id}_X\) for all \(f\). Then \begin{align*} \lim_I F({-}) \coloneqq\mathop{\mathrm{Mor}}_{\mathsf{Cat}}(\chi_{({-})}, F): \mathsf{C} &\to {\mathsf{Set}}\\ \end{align*} i.e. the limit is the functor sending \(X\) to natural transformations between \(\chi_X\) and \(F\). If this functor is representable, it can be identified with an object in \(\mathsf{C}\).

Filtered colimits

Examples

• \(\lim(\bullet \rightrightarrows\bullet)\) is an equalizer
• For \(I\) an index set regarded as a groupoid with only identities, \(\lim(\bullet, \bullet, \cdots) = \prod_{i\in I} \bullet_i\) . Use that \(\lim_I F(X) = \left\{{X\to X_i}\right\}\) if \(F(i) \coloneqq X_i\).
• \(\lim(\bullet \to \bullet \leftarrow\bullet) = \bullet{ \underset{\scriptscriptstyle {\bullet} }{\times} }\bullet\) is a fiber product.

Limits

Build new objects by “imposing equations” on existing ones. - Ex: Construction of the Unsorted/p-adic as the limit of the sequence of quotient homomorphisms: \(\cdots \rightarrow \mathbb{Z}/p^n \rightarrow \cdots \rightarrow \mathbb{Z}/p^2 \rightarrow \mathbb{Z}/p\)

Filtered categories and colimits