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Limits (Categorical)
Modern POV: realize in terms of a Kan extension: 
Mnemonics to distinguish lims and colims
Inverse/projective limit Direct/colimit
\(\lim\) \(\colim\)
Above: 
Below: 
terminal cones initial cocones
subobjects of \(\displaystyle\prod\) quotient objects of \(\coprod\) or \(\bigoplus\)
terminal objects initial objects
pullbacks pushouts
products coproducts
kernels cokernels
equalizers coequalizers
commutes with contravariant hom: \(\mathsf{C}(A, \lim B_i) = \lim \mathsf{C}(A, B_i)\) commutes with covariant hom: \(\mathsf{C}(\lim A_i, B) \cong \colim \mathsf{C}(A_i, B)\)
continuous cocontinuous
“impose equations” “glue objects”
\({ {\mathbf{Z}}_{\widehat{p}} }= \lim(\cdots \xrightarrow{\operatorname{mod}p^2} {\mathbf{Z}}/p^2{\mathbf{Z}}\xrightarrow{\operatorname{mod}p} {\mathbf{Z}}/p{\mathbf{Z}})\) \({\mathbf{Z}}(p^\infty) = \colim({\mathbf{Z}}/p{\mathbf{Z}}\xrightarrow{\times p} {\mathbf{Z}}/p^2{\mathbf{Z}}\xrightarrow{\times p} \cdots)\)
Idea: the limit should be the closest object to all of the \(F(X_i)\).
The Calculus of Limits
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Commuting lims:
- Self-commuting: lims commute with lims, colims commute with colims.
- There is a morphism \(\colim \lim A_{i, j} \to \lim \colim A_{i, j}\) which need not be an isomorphism, i.e. limits need not commute with colimits in general.
- Filtered colimits commute with finite limits in \({\mathsf{Set}}\).
-
Preservation of properties:
- Limits respect finite group actions, i.e. \((\lim A_i)/G { \, \xrightarrow{\sim}\, }\lim (A_i/G)\).
- Finite colims of compact objects are compact
-
The acronyms for adjoints:
- RAPL: right adjoints preserve limits
- LAPC: left adjoints preserve colimits
- LARE: Left adjoints are right-exact
- RALE: right adjoints are left-exact
Definitions
Limits
Colimits
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{\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 \(\mathsf{C}\).
Definition in terms of cones and cocones:
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.
Filtered categories and colimits
