Category theory; homological algebra Definition

Inverse limit

Builds on Directed set Category Morphism Limit

Definition

Let $(I, \le)$ be a directed set and let $\mathcal{C}$ be a category. An inverse system in $\mathcal{C}$ over $I$ is a family of objects $A_i$ for $i \in I$ with morphisms $f_{ij} \colon A_j \to A_i$ for all $i \le j$ , such that:

$f_{ii} = \id_{A_i}$ for all $i$ .
$f_{ij} \circ f_{jk} = f_{ik}$ for all $i \le j \le k$ .

An inverse limit of this system is an object $A$ with morphisms $\pi_i \colon A \to A_i$ such that:

$f_{ij} \circ \pi_j = \pi_i$ for all $i \le j$ .
For every object $B$ with morphisms $g_i \colon B \to A_i$ such that $f_{ij} \circ g_j = g_i$ for all $i \le j$ , there is a unique morphism $u \colon B \to A$ with $\pi_i \circ u = g_i$ for all $i$ .

Remarks

The inverse limit is written $\varprojlim A_i$ . It is the limit of the diagram given by the system, and it is unique up to a unique isomorphism. In sets, groups, rings and modules it always exists. There it is the set of families $(a_i) \in \prod_i A_i$ with $f_{ij}(a_j) = a_i$ for all $i \le j$ , with the projections as $\pi_i$ .

Example: the rings $\Z/p^n\Z$ for $n \ge 1$ , with the reduction maps, form an inverse system over $\N$ . Its inverse limit is the ring $\Z_p$ of $p$ -adic integers. Clause 1 holds because reducing a compatible sequence at level $n+1$ gives its entry at level $n$ .

Non-example: $\Z$ with the reduction maps $\Z \to \Z/p^n\Z$ satisfies clause 1 but not clause 2. Taking $B = \Z_p$ , a morphism $u \colon \Z_p \to \Z$ with $\pi_n \circ u$ equal to reduction would have to be injective, which is impossible since $\Z_p$ is uncountable.

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Category concept
Directed set concept
Limit concept
Morphism concept
Inverse limit definition

Tag	`XAQSNBJSX4`
URI	`m2://XAQSNBJSX4`
Kind	definition
Writing	W0 stub
MSC	18A30
Defines	Inverse limit
Uses	Directed set, Category, Morphism, Limit

Inverse limit

Definition

Remarks

Reading path

Further reading

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