Contents

category theory

# Contents

## Idea

A weak limit for a diagram in a category is a cone over that diagram which satisfies the existence property of a limit but not necessarily the uniqueness. The dual concept is a weak colimit.

Beware that “weak” here does not correspond to that in “weak n-category”, in particular it does not refer to homotopy limits. Nevertheless, there is a relation, see below. It is due to this relation that weak limits in homotopy categories play a key role in the Brown representability theorem.

## Weak pullbacks

A weak pullback of a cospan

$A\overset{f}{\to} C \overset{g}{\leftarrow} B$

(in some category) is a commutative square

$\array{ P & \overset{p}{\to} & A \\ {}^{\mathllap{q}} \downarrow && \downarrow^{\mathrlap{f}} \\ B & \overset{g}{\to} & C }$

such that for every commuting square

$\array{ X & \overset{x}{\to} & A\\ {}^{\mathllap{y}} \downarrow && \downarrow^{\mathrlap{f}}\\ B & \overset{g}{\to} & C}$

there exists a morphism $h: \colon X\to P$, not necessarily unique, such that $x = h p$ and $y = h q$;

If the actual pullback $A \underset{C}{\times}B$ exists, then this condition means equivalently that the universal morphism

$P \longrightarrow A \underset{C}{\times}B$

is a split epimorphism.

## Weak terminal objects

Every inhabited set is a weak terminal object in Set, since there always exists a function from any set to any inhabited set. But only a singleton is a terminal object.

## Projective objects and exact completion

In any category with finite limits and enough projectives, the full subcategory of projective objects has weak finite limits. For example, given a cospan $A\overset{f}{\to} C \overset{g}{\leftarrow} B$ of projective objects, let $P\to A\times_C B$ be a projective cover of the actual pullback; then any square

$\array{ X & \overset{x}{\to} & A\\ ^y \downarrow && \downarrow ^f\\ B & \overset{g}{\to} & C}$

with $X$ projective induces a morphism $X\to A\times_C B$, which lifts to a morphism $X\to P$ since $X$ is projective.

Conversely, from any category with weak finite limits one can construct an exact completion in which the original category sits as the projective objects, and the exact categories constructible in this way are precisely those having enough projectives.

## Relation to homotopy limits

Unlike usages of ‘weak’ in terms like weak n-category, a weak limit is not be like a homotopy limit or a 2-limit, which satisfy uniqueness (as well as existence) albeit only up to higher homotopies or equivalences.

However, some homotopy limits induce the corresponding type of weak limit in the corresponding homotopy category. For example, suppose that

$\array{ P & \overset{p}{\to} & A\\ ^q \downarrow && \downarrow ^f\\ B & \overset{g}{\to} & C}$

is a homotopy pullback in some category $M$ having a notion of homotopy, such as a model category. In particular, this square commutes up to homotopy, and thus it commutes in the homotopy category $Ho(M)$. Then any square

$\array{ X & \overset{x}{\to} & A\\ ^y \downarrow && \downarrow ^f\\ B & \overset{g}{\to} & C}$

that commutes in $Ho(M)$ commutes up to homotopy in $M$, and thus (by the (“local”) universal property of homotopy pullbacks), there is a map $h:X\to P$ and homotopies $p h \simeq x$ and $q h\simeq y$; thus the given square is a weak pullback in $Ho(M)$. While the universal property of a homotopy pullback means that $h$ is unique up to homotopy, this is only true for a given choice of homotopy $f x \simeq g y$, and different such homotopies can induce inequivalent $h$‘s. Thus in $Ho(M)$, which remembers only the existence of homotopies, we have only a weak pullback.

Note, though, that not all homotopy limits produce weak limits in the homotopy category, because in general it will not be possible to lift a cone that commutes in $Ho(M)$ to a cone that commutes up to coherent homotopy in $M$. However, in “simple” cases such as pullbacks, products, equalizers, sequential inverse limits, and so on, this is always true (and it will be true whenever the diagram category is a quiver). On the other hand, homotopy products in $M$ give actual (not weak) products in $Ho(M)$, since there are no homotopies necessary.

• Peter Freyd, Representations in abelian categories 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 95–120 Springer, New York