nLab
created limit

Creation of limits

Creation of limits

Definition

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

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

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

Examples

A monadic functor creates all limits that exist in its codomain, and all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself). 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 “FF creates limits for JJ if JJ has a limit whenever FJF\circ J has a limit, and in that case limits of JJ are preserved and reflected by FF.” (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 “FF creates limits for JJ if JJ has a limit whenever FJF\circ J has a limit, and furthermore in all cases limits of JJ are preserved and reflected by FF.”

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 “FF creates limits for JJ if JJ and FJF\circ J both have limits, and furthermore limits of JJ are preserved and reflected by FF.”

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 LL over FJF J in DD there exists a unique cone LL' over JJ which is mapped exactly to LL, and this LL' is a limit of JJ. 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.

Remarks

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

Last revised on February 4, 2018 at 05:10:06. See the history of this page for a list of all contributions to it.