nLab created limit

Definition

Let $F\colon C\to D$ be a functor and $J\colon I\to C$ a diagram, and suppose that the composite $F \circ J$ has a limit. We say that $F$ creates this limit if $J$ has a limit, and $F$ both preserves and reflects limits of $J$. The latter two conditions together mean that a cone over $J$ in $C$ is a limiting cone if and only if its image in $D$ is a limiting cone over $F\circ J$.

Of course, a functor $F$ creates a colimit if $F^{op}$ creates the corresponding limit.

If $F$ creates all limits or colimits of a given type (i.e. over a given category $I$), we simply say that $F$ creates that sort of limit (e.g. $F$ creates products, $F$ creates equalizers, etc.).

Examples

Proposition

(monadic functors create limits) A monadic functor

1. creates all limits that exist in its codomain;

2. creates all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself).

(e.g. MacLane 71, Exercise IV.2.2 (p. 138))

Creation of a particular sort of split coequalizer figures prominently in Beck’s monadicity theorem.

Terminological remarks

Creation of non-existing limits

It seems that the notion of “creating a limit” is used most frequently when the limits exist in the codomain. One may want to extend the terminology to cases when such limits don’t exist, which would require making a choice about whether a non-existing limit should be regarded as “created”.

In Categories Work the convention is that a functor creates all limits that do not exist in its codomain. In this case, the more generally applicable definition could be stated as “$F$ creates limits for $J$ if $J$ has a limit whenever $F\circ J$ has a limit, and in that case limits of $J$ are preserved and reflected by $F$.” (But see below for an additional difference with Categories Work.)

On the other hand, one might argue that it doesn’t make sense to regard a limit that exists in the domain as being “created by the functor” if the limit in the codomain doesn’t even exist. In this case the more generally applicable definition could be stated as “$F$ creates limits for $J$ if $J$ has a limit whenever $F\circ J$ has a limit, and furthermore in all cases limits of $J$ are preserved and reflected by $F$.”

Finally, one might even argue that based on the meaning of the English word “created”, only something that exists can be created at all. In this case the more generally applicable definition could be stated as “$F$ creates limits for $J$ if $J$ and $F\circ J$ both have limits, and furthermore limits of $J$ are preserved and reflected by $F$.”

Strictness

The definitions given above are all “up to isomorphism”, i.e. they satisfy the principle of equivalence. The definition in Categories Work is additionally strict: it requires that for every limiting cone $L$ over $F J$ in $D$ there exists a unique cone $L'$ over $J$ which is mapped exactly to $L$, and this $L'$ is a limit of $J$. This is used in stating the version of the monadicity theorem that characterizes the category of algebras for a monad up to isomorphism rather than equivalence of categories. For amnestic isofibrations the strict and the non-strict notion are equivalent.

Remarks

Kissinger suggested a concise way to state creation/preservation/etc. of limits. However, there is some dispute about its correctness.

Related pages

• preserved limit

• reflected limit

• created limit

• lifted limit

References

• Saunders Mac Lane, Definition V.1 in: Categories for the Working Mathematician (1971)

• Jiří Adámek, Horst Herrlich, George E. Strecker, Definition 13.17(2) in: Abstract and Concrete Categories.

• Emily Riehl, §3.3 in: Category Theory in Context, Dover Publications (2017) [pdf, book website]