Category theory; homological algebra Definition

Filtered category

Definition

Let $J$ be a category. $J$ is filtered if:

$J$ has at least one object.
For any two objects $a$ and $b$ of $J$ , there are an object $c$ and morphisms $a \to c$ and $b \to c$ .
For any two morphisms $f, g \colon a \to b$ of $J$ , there is a morphism $h \colon b \to c$ with $h \circ f = h \circ g$ .

Remarks

Equivalently, every diagram $F \colon K \to J$ with $K$ a finite category has a cocone. A colimit over a filtered category is a filtered colimit. The dual notion is a cofiltered category.

Example: a directed set $(I, \le)$ , seen as a category with one arrow $i \to j$ when $i \le j$ , is filtered. Clause 1 holds because a directed set is not empty, and clause 2 is the existence of a common upper bound. Clause 3 holds because there is at most one arrow between two objects, so $f = g$ .

Non-example: the category with two objects $a, b$ , two arrows $f, g \colon a \to b$ and only identities besides, is not filtered. Clause 3 fails: the only arrow out of $b$ is $\id_b$ , and $\id_b \circ f \neq \id_b \circ g$ .

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Category concept
Morphism concept
Filtered category definition

Tag	`TKG3YF2G5A`
URI	`m2://TKG3YF2G5A`
Kind	definition
Writing	W0 stub
MSC	18A30
Defines	Filtered category
Uses	Category, Morphism

Filtered category

Definition

Remarks

Reading path

Further reading

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