Examples:: [[equalizer]] Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* ---- > [!definition] Definition. ([[categorical limit]]) > Let $\mathscr{F}: \mathsf{I} \to \mathsf{ C}$ be a [[covariant functor]], where one thinks of $\mathsf{I}$ as a [[category]] 'of indices'. The **limit** of $\mathscr{F}$ is (if it exists) an object $L \in \text{Obj}(\mathsf{C})$, endowed with morphisms $\lambda_{I}: L \to \mathscr{F}(I)$ for all objects $I \in \text{Obj}(\mathsf{I})$, satisfying the following properties: > > - If $\alpha: I \to J$ is a morphism in $\mathsf{I}$, then $\lambda_{J}=\mathscr{F}(\alpha) \circ \lambda_{I}$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \usepackage[mathscr]{euscript} > > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAZEAX1PU1z5CKAEyli1Ok1bsAOrIC29HAAs4AYwBOwAGLcAFACkAlDz4gM2PASLlxkhizaIQ8pao3a9+gJKnukjBQAObwRKAAZpoQCkh2IDgQSGRSTnKyjPQKAEZQ9AD6hmaR0bGI8YlIYqkyLvKZOXn5PiA0mdkwjAAKAtbCIJpYwSo4xSBRMVU0lYgpjrWuispqWroG8kxoKvSmbfQd3b1C7IPDowHcQA > \begin{tikzcd} > & L \arrow[rd, "\lambda_J"] \arrow[ld, "\lambda_I"'] & \\ > \mathscr{F}(I) \arrow[rr, "\mathscr{F}(\alpha)"'] & & \mathscr{F}(J) > \end{tikzcd} > \end{document} > ``` > > - $L$ is [[terminal object|final]] with respect to this property of factoring through $\mathscr{F}(I)$ via $\mathscr{F}(\alpha)$: that is, if $M$ is another object, endowed with morphisms $\mu_{I}$, also satisfying the previous requirement, then there exists a unique morphism $M \to L$ making all relevant diagrams commute. > > The limit $L$ (if it exists) is unique up to isomorphism, [[terminal objects are unique up to a unique isomorphism|as is every notion defined by a university property]]. It is denoted $\lim\limits_{{\longleftarrow}} \mathscr{F}$. The left-arrow reminds that $L$ stands 'before' all objects of $\mathsf{C}$ indexed by $\mathsf{I}$ via $\mathscr{F}$ (but it is—up to isomorphisms—the 'last' object that may do so). Suppose $\mathsf{I}$ is a ~~[[filtered category]] (e.g. a [[filtered poset|directed set]])~~ [[poset]]. Any [[diagram]] $h$ in $\mathsf{Set}$ indexed by $\mathsf{I}^{\text{op}}$ (an **inverse system**) has the following, together with the obvious (restricted) projections, as an inverse limit: (something like this) $L=\left\{ \boldsymbol y = (y_{i})_{i \in I}\in \prod_{i \in I} h(i) : \forall i \leq j, h_{ij}(y_{j})=y_{i} \right\}.$ > [!example] Example. (Products) > Let $\mathsf{I}$ be the 'discrete category' consisting of two objects $\boldsymbol 1, \boldsymbol 2$, with only identity morphisms, and let $\mathscr{A}$ be a [[covariant functor|functor]] from $\mathsf{I}$ to any [[category]] $\mathsf{C}$. Let $A_{1}=\mathscr{A}(\boldsymbol 1)$ and $A_{2}=\mathscr{A}(\boldsymbol 2)$ be the two objects of $\mathsf{C}$ 'indexed' by $\mathsf{I}$. In this case, no $\alpha$ (in the sense from the definition) exists... we are left with the following diagram characterizing $L$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoAmAXVJADcBDAGwFcYkQBBAfQEYQBfUuky58hFOQrU6TVu27kBQkBmx4CRHqR7SGLNohAAZJcLVjNpYrtkGQAWQHSYUAObwioAGYAnCAFskSRAcCCRiQW8-QMRg0KQeSJBfAKQAZhp4xDIQACMYMCgkAFo0iOUUmIyQsMQtPIKixDKkyvTM2uDGLDA7KHo4AAsXEBo9OUMAHUmYAA8sOBw4AAIAQid+IA > \begin{tikzcd} > & M \arrow[ldd, bend right] \arrow[rdd, bend left] \arrow[d, "\exists !", dashed] & \\ > & L \arrow[ld] \arrow[rd] & \\ > A_1 & & A_2 > \end{tikzcd} > \end{document} > ``` > But this is just the [[universal property]] defining [[categorical product|products]] (e.g. see [[universal property of product sets]]). Thus, a limit exists iff a product of $A_{1}$ and $A_{2}$ exists in $\mathsf{C}$. More generally, one may consider the notion of a [[categorical pullback|pullback]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```