nLab
2-limit

Idea

A 2-limit is the type of limit that is appropriate in a (weak) 2-category. There are three notable changes when passing from ordinary 1-limits to 2-limits:

1. Since in a 2-category we avoid evil, the “cones” in a 2-limit are required to commute only up to isomorphism.

2. The universal property of the limit is expressed by an equivalence of categories rather than a bijection of sets. This means that (1) every other cone over the diagram that commutes up to isomorphism factors through the limit, up to isomorphism, and (2) every transformation between cones also factors through a 2-cell in the limit. We will give some examples below.

3. Since 2-categories are enriched over Cat (this is precise in the strict case, and weakly true otherwise), Cat-weighted limits become important. This means that both the diagrams we take limits of and the shape of “cones” that limits represent can involve 2-cells as well as 1-cells.

$2$-colimits

Everything below applies dually to $2$-colimits, the higher analogues of colimits. (But somebody might want to make a separate page that gives appropriate examples of these.)

Strictness and terminology

In a strict 2-category, one can also consider strict 2-limits, which are precisely Cat-weighted limits in the sense of ordinary enriched category theory. In a strict 2-category these “strict” limits are often technically useful in constructing the “up-to-isomorphism” 2-limits we consider here. See strict 2-limit for details.

When we know we are working in a (weak) 2-category, the only type of limit that makes sense is a 2-limit. Therefore, we usually call these simply “limits.” To emphasize the distinction with the strict 2-limits in a strict 2-category, the “up-to-isomorphism” 2-limits were historically often called bilimits (by analogy with bicategory for “weak 2-category”). However, this terminology is somewhat unfortunate, not only because it doesn’t generalize well to $n$, but because it leads to words like “biproduct,” which also has the completely unrelated meaning of an object that is both a product and a coproduct (which is common in additive categories).

Unfortunately, we probably shouldn’t use “weak limit” to emphasize the “up-to-isomorphism” nature of these limits, because that also has the completely unrelated meaning of an object in a 1-category satisfying the existence, but not the uniqueness property of an ordinary limit.

Examples

Any ordinary type of limit can be “2-ified” by boosting its ordinary universal property up to a 2-categorical one. In the following examples we work in a 2-category $K$.

• A terminal object in $K$ is an object 1 such that $K(X\mathrm{,1})$ is equivalent to the terminal category for any object $X$. This means that for any $X$ there is a morphism $X\to 1$ and for any two morphisms $f,g:X\to 1$ there is a unique isomorphism $f\cong g$.

• A product of two objects $A,B$ in $K$ is an object $A\times B$ together with a natural equivalence of categories $K(X,A\times B)\simeq K(X,A)\times K(X,B)$. This means that we have projections $p:A\times B\to A$ and $q:A\times B\to B$ such that (1) for any $f:X\to A$ and $g:X\to B$, there exists an $h:X\to A\times B$ and isomorphisms $ph\cong f$ and $qh\cong g$, and (2) for any $h,k:X\to A\times B$ and 2-cells $\alpha :ph\to pk$ and $\beta :qh\to qk$, there exists a unique $\gamma :h\to k$ such that $p\gamma =\alpha$ and $q\gamma =\beta$.

• A pullback of a co-span $A\stackrel{f}{\to }C\stackrel{g}{\leftarrow }B$ consists of an object $A{\times }_{C}B$ and projections $p:A{\times }_{C}B\to A$ and $q:A{\times }_{C}B\to B$ together with an isomorphism $\varphi :fp\cong gq$, such that (1) for any $m:X\to A$ and $n:X\to B$ with an isomorphism $\psi :fm\cong gn$, there exists an $h:X\to A{\times }_{C}B$ and isomorphisms $\alpha :ph\cong m$ and $\beta :qh\cong n$ such that $g\beta .\varphi h.f\alpha =\psi$, and (2) a suitable condition on 2-cells as well. This is sometimes called the pseudo-pullback but that term more properly refers to a particular strict 2-limit.

• An equalizer of $f,g:A\to B$ consists of an object $E$ and a morphism $e:E\to A$ together with an isomorphism $fe\cong ge$, which is universal in a sense the reader should now be able to write down. This is sometimes called the pseudo-equalizer but that term more properly refers to a particular strict 2-limit. Note that frequently, such as in the construction of all limits from basic ones, equalizers need to be replaced by descent object?s.

There are also various important types of 2-limits that involve 2-cells in the diagram shape or in the weight, and hence are not just “boosted-up” versions of 1-limits.

• The comma object of a cospan $A\stackrel{f}{\to }C\stackrel{g}{\leftarrow }B$ is a universal object $(f/g)$ and projections $p:(f/g)\to A$ and $q:(f/g)\to B$ together with a transformation (not an isomorphism) $fp\to gq$. In Cat, comma objects are comma categories. Comma objects are sometimes called lax pullbacks but that term more properly refers to the lax version of a pullback; see “lax limits” below.

• The inserter? of a pair of parallel arrows $f,g:A\rightrightarrows B$ is a universal object $I$ equipped with a map $i:I\to A$ and a 2-cell $fi\to gi$.

• The equifier? of a pair of parallel 2-cells $\alpha ,\beta :f\to g:A\to B$ is a universal object $E$ equipped with a map $e:E\to A$ such that $\alpha e=\beta e$.

• The inverter? of a 2-cell $\alpha :f\to g:A\to B$ is a universal object $V$ with a map $v:V\to A$ such that $\alpha v$ is invertible.

• The power of an object $A$ by a category $C$ is a universal object $A^C$ equipped with a functor $C\to K(A^C,A)$. Of particular importance is the case when $C$ is the walking arrow $2$.

Lax limits

A lax limit is like a 2-limit, except that the triangles in the definition of a cone are required only to commute up to a specified transformation, not necessarily an isomorphism. The lax limit of one diagram can always be converted to an ordinary 2-limit by changing the weight.

• Lax terminal objects and lax products are the same as ordinary ones, since there are no commutativity conditions on the cones.

• The lax limit of an arrow $f:A\to B$ is a universal object $L$ equipped with projections $p:L\to A$ and $q:L\to B$ and a 2-cell $fp\to q$. Note that this is equivalent to a comma object $(f/1_B)$.

• The lax pullback of a cospan $A\stackrel{f}{\to }C\stackrel{g}{\leftarrow }B$ is a universal object $P$ equipped with projections $p:P\to A$, $q:P\to B$, $r:P\to C$, and 2-cells $fp\to r$ and $gq\to r$.

Note that lax pullbacks are not the same as comma objects. In general comma objects are much more useful, but there are 2-categories that admit all lax limits but do not admit comma objects, so using “lax pullback” to mean “comma object” can be misleading.

Finite Limits

A 2-limit is called finite if its diagram shape and its weight are both “finitely presentable” in a suitable sense. Pullbacks, comma objects, inserters, equifiers, and so on are all finite limits, as are powers by any finitely presented category. All finite limits can be constructed from pullbacks, a terminal object, and powers with $2$.

References

• Street, “Limits indexed by category-valued 2-functors”

• Kelly, “Elementary observations on 2-categorical limits”

• Street, “Fibrations in Bicategories” and correction.

• Lack, A 2-categories companion