Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
This entry lists and discusses examples and special types of limits and colimits, hence also in particular of products and coproducts.
It starts with very elementary and simple examples and eventually passes to more sophisticated ones.
For examples of the other universal constructions see
Examples of limits
In the following examples, is a small category, is any category and the limit is taken over a functor .
the limit of the empty diagram in is, if it exists the terminal object;
if is a discrete category, i.e. a category with only identity morphisms, then a diagram is just a collection of objects of . Its limit is the product of these.
if then is the equalizer of the two morphisms .
if has an terminal object (so that is an initial object in ), then the limit of any is .
if is a poset, then the limit over is the supremum over the with respect to ;
the generalization of this is where the term “limit” for categorical limit (probably) originates from: for a filtered category, hence a cofiltered category, one may think of as witnessing that is “larger than” in some sense, and is then the “largest” of all these objects, the limiting object. This interpretation is perhaps more evident for filtered colimits, where the codomain category is thought of as being the opposite . See the motivation at ind-object.
In terms of other operations
If products and equalizers exist in , then limit of can be exhibited as a subobject of the product of the , namely the equalizer of
See the explicit formula for the limit in Set in terms of a subset of a product set.
In particular therefore, a category has all limits already if it has all products and equalizers.
Limits and colimits in Set
In the category Set of sets, limits and colimits reduce to the very familiar operations of
Conversely, limits and colimits in other categories may be regarded as generalizations of these concepts to things other than plain sets.
the limit over any is – this is equivalently
therefore for every set , there is a natural bijection , where on the right the limit is taken of the functor .
the limit over a Set-valued functor is a subset of the product of all objects: .
the colimit over a Set-valued functor is a quotient set of the disjoint union :
where the equivalence relation is that which is generated by
If is a filtered category then the relation already is an equivalence relation.
Limits in presheaf categories
Consider limits of functors with values in the category of presheaves on a small category .
Limits of presheaves are computed objectwise:
Here on the right the limit is over the functor .
Similarly colimits of presheaves are computed objectwise.
The Yoneda embedding commutes with small limit:
Let , then we have
Warning The Yoneda embedding does not in general preserve colimits.
Limits in under-categories
Limits in under categories are a special case of limits in comma categories. These are explained elsewhere. It may still be useful to spell out some details for the special case of under-categories. This is what the following does.
Limits in an under category are computed as limits in the underlying category.
Precisely: let be a category, an object, and the corresponding under category, and the obvious projection.
Let be any functor. Then, if it exists, the limit over in is the image under of the limit over :
and is uniquely characterized by .
Over a morphism in the limiting cone over (which exists by assumption) looks like
By the universal property of the limit this has a unique lift to a cone in the under category over :
It therefore remains to show that this is indeed a limiting cone over . Again, this is immediate from the universal property of the limit in . For let be another cone over in , then is another cone over in and we get in a universal morphism
A glance at the diagram above shows that the composite constitutes a morphism of cones in into the limiting cone over . Hence it must equal our morphism , by the universal property of , and hence the above diagram does commute as indicated.
This shows that the morphism which was the unique one giving a cone morphism on does lift to a cone morphism in , which is then necessarily unique, too. This demonstrates the required universal property of and thus identifies it with .
- Remark: One often says “ reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if is monadic (i.e., has a left adjoint such that the canonical comparison functor is an equivalence), then both reflects and preserves limits. In the present case, the projection is monadic, is essentially the category of algebras for the monad , at least if admits binary coproducts. (Added later: the proof is even simpler: if is the underlying functor for the category of algebras of an endofunctor on (as opposed to algebras of a monad), then reflects and preserves limits; then apply this to the endofunctor above.)
Pedagogical vidoes that explain limits and colimits are at
A web-based program that generates componentwise illustrations of simple limits and colimits in Set is developed at
More on the inner workings of this program is at Paine on a Category Theory Demonstrations program
the following discussion originated from an earler version of this entry
Todd Trimble: So far, this is a really good article. However, I would not say in this last line “if either limit exists”, because small limits on the right certainly exist always since is complete; instead, “if exists”.
Urs: thanks, Todd, I have changed the above now accordingly. Please don’t hesitate to correct and/or improve things you see as needed.
By the way, I am not completely happy with this entry as yet. It was originally motivated from the desire to explain in small steps the computation of limits and colimits to those readers unfamiliar with it. Currently this here mostly just lists results, where maybe we would eventually want to include also pedagocial proofs.
The material below “explanation for programmers” goes more in that pedagogical direction, though I’d think eventually it would be good to also have the kind of pedestrian explanation given there but without (at first) its realization in Python! :-)
an earlier version of this entry, which contained the material now branched off at Paine on a Category Theory Demonstrations program, led to the following discussion
Urs Schreiber: sorry to say this, but I am not so happy with the following material here at this particular entry. This entry here is supposed to explain limits and colimits. Originally I thought that the computer program described below should be used here to help explain limits and colimits. For instance by using its graphical output for illustration purposes. But instead the material below explains how to write that program . That may be of interest, too, but here at this entry it seems a bit of a distraction. Could we move the following material to its seperate entry?
Toby: I would agree that the material on how to write the program would work well in a separate entry, say programming coproducts?. On the other hand, you definitely want to keep the first two lines here; they do just what you want and could be expanded on here.