# Projective limit

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inverse limit

A construction that arose originally in set theory and topology, and then found numerous applications in many areas of mathematics. A common example of a projective limit is that of a family of mathematical structures of the same type indexed by the elements of a pre-ordered set. Let $I$ be a set endowed with a pre-order relation $\leq$, and suppose that with each element $i \in I$ a set $X _ {i}$ is associated and with each pair $( i , j )$, $i , j \in I$, in which $i \leq j$, a mapping $\phi _ {ij} : X _ {i} \rightarrow X _ {j}$ is associated, where the $\phi _ {ii}$, $i \in I$, are identity mappings and $\phi _ {jk} \phi _ {ij} = \phi _ {ik}$ for $i \leq j \leq k$. A set $X$ is called a projective limit of the family of sets $X _ {i}$ and mappings $\phi _ {ij}$ if the following conditions are satisfied: a) there exists a family of mappings $\pi _ {i} : X \rightarrow X _ {i}$ such that $\phi _ {ij} \pi _ {i} = \pi _ {j}$ for any pair $i \leq j$; b) for any family of mappings $\alpha _ {i} : Y \rightarrow X _ {i}$, $i \in I$, from an arbitrary set $Y$, for which the equalities $\phi _ {ij} \alpha _ {i} = \alpha _ {j}$ hold for $i \leq j$, there exists a unique mapping $\alpha : Y \rightarrow X$ such that $\alpha _ {i} = \pi _ {i} \alpha$ for every $i \in I$. The projective limit can be described explicitly as follows. One considers the direct product $\prod _ {i \in I } X _ {i}$ and selects in it the set of all functions $f : I \rightarrow \cup _ {i \in I } X _ {i}$ for which the equalities $\phi _ {ij} ( f ( i) ) = f ( j)$ hold for $i \leq j$. This subset is the projective limit of the family $X _ {i}$. If all the $X _ {i}$ are equipped with an additional structure of the same type and the $\phi _ {ij}$ preserve it, then the same structure is induced in the projective limit. Therefore it is possible to speak of projective limits of groups, modules, topological spaces, etc.

A natural generalization of this concept of projective limit is that of the projective limit of a functor. Let $F : \mathfrak D \rightarrow \mathfrak K$ be a functor from a small category $\mathfrak D$ to an arbitrary category $\mathfrak K$. An object $X \in \mathop{\rm Ob} \mathfrak K$ together with morphisms $\pi _ {D} : X \rightarrow F ( D)$, $D \in \mathop{\rm Ob} \mathfrak D$, is called a projective limit (inverse limit, or simply limit) of the functor $F$ if the following conditions are satisfied: $\alpha$) $F ( \phi ) \pi _ {D} = \pi _ {D ^ \prime }$ for any morphism $\phi : D \rightarrow D ^ \prime$; and $\beta$) for any family of morphisms $\alpha _ {D} : Y \rightarrow F ( D)$ satisfying $F ( \phi ) \alpha _ {D} = \alpha _ {D ^ \prime }$ for all $\phi : D \rightarrow D ^ \prime$ there exists a unique morphism $\alpha : Y \rightarrow X$ such that $\alpha _ {D} = \phi _ {D ^ \prime } \alpha$ for all $D \in \mathop{\rm Ob} \mathfrak D$. Notation: $\lim\limits F = ( X , \pi _ {D} )$.

### Examples of projective limits.

1) Let $I$ be a discrete category. Then for an arbitrary functor $F : I \rightarrow \mathfrak K$ the projective limit of the functor $F$ coincides with the product of the family of objects $F ( i)$, $i \in I$( cf. Product of a family of objects in a category).

2) Let $\mathfrak D$ be a category with two objects $A , B$ and two non-identity morphisms $\alpha , \beta : A \rightarrow B$. Then the limit of a functor $F : \mathfrak D \rightarrow \mathfrak K$ is the equalizer of the pair of morphisms $F ( \alpha ) , F ( \beta )$( cf. Kernel of a morphism in a category).

If a category has products of arbitrary small families of objects and equalizers of pairs of morphisms, then it has limits for all functors defined on small categories.