nLab
connected limit

Definition

A connected limit is a limit over a connected category. Similarly, a connected colimit is a colimit over a connected category.

Examples

The following are all connected limits:

• pullbacks

• wide pullbacks

• equalizers

• cofiltered limits

• split idempotents

Properties

Construction from pullbacks and equalizers

Theorem

If a category $C$ has pullbacks and equalizers, then it has all finite connected limits.

Proof

Let $I$ be a finite connected category and $F:I\to C$ a functor. Since $I$ is connected, it is inhabited; choose some object $x_0\in I$. For each object $x\in I$, let $\ell (x)$ be the length of the shortest zigzag from $x_0$ to $x$. Now order the objects of $I$ as $x_0,x_1,\dots ,x_n$ such that for all $i$ we have $\ell (x_i)\le \ell (x_{i+i})$.

Now we inductively define objects $P_i\in C$, for $0\le i\le n$, with projections $p_{ij}:P_i\to F(x_j)$ for $j\le i$. We begin with $P_0=x_0$ and $p_{00}=1_{x_0}$. Assuming $P_i$ and $p_{ij}$ defined, choose a zigzag from $x_0$ to $x_{i+1}$ of minimal length, say

$$x_0\leftrightarrow y_1\leftrightarrow \dots \leftrightarrow y_k\leftrightarrow x_{i+1}.$$x_0 \leftrightarrow y_1 \leftrightarrow \dots \leftrightarrow y_k \leftrightarrow x_{i+1}.

By our choice of the ordering of the objects of $I$, we have $y_k=x_j$ for some $j\le i$, and thus we have $q=p_{ij}:P_i\to y_k$.

If the final arrow $y_k\leftrightarrow x_{i+1}$ in the zigzag is directed as $y_k\to x_{i+1}$, then let $P_{i+1}=P_i$, let $p_{i+1,i+1}$ be the composite $P_i\stackrel{q}{\to }F(y_k)\to F(x_{i+1})$, and keep the other $p_{ij}$ unchanged. On the other hand, if $y_k\leftrightarrow x_{i+1}$ is directed as $y_k\leftarrow x_{i+1}$, let $P_{i+1}$ be the pullback

$$\begin{array}{cccc}& P_{i+1}& \to & P_i\\ {}^{p_{i+1,i+1}}& \downarrow & & {\downarrow }^q\\ & F(x_{i+1})& \to & F(y_k)\end{array}$$\array{ & P_{i+1} & \to & P_i \\ ^{p_{i+1,i+1}} & \downarrow & & \downarrow^q \\ & F(x_{i+1}) & \to & F(y_k) }

and define $p_{i+1,j}$ for $j\le i$ by composition with $P_{i+1}\to P_i$.

At the end of this procedure, we have an object $P_n$ with projections $p_{n,j}:P_n\to F(x_j)$ for all objects $x_j\in I$. Now we order the morphisms in $I$ as $g_0,\dots ,g_m$ and define, inductively, an object $Q_i$ with a morphism $q_i:Q_i\to P_n$. We begin with $Q_0=P_n$ and $q_0=1_{P_n}$. Given $Q_i$ and $q_i$, we let $e:Q_{i+1}\to Q_i$ be the equalizer of $F(g_{i+1})\circ p_{n,?}\circ q_i$ and $p_{n,?}\circ q_i$, where the ?s denote the indices of the objects that are the source and target of $g_{i+1}$. We then set $q_{i+1}=q_i\circ e$.

At the end of this procedure, we have an object $Q_{m+1}$ together with a cone over the diagram $F$, which is easily verified to be a limit of $F$.

Similarly, arbitrary connected limits may be built from wide pullbacks and equalizers.

Theorem

Let $C$ be (finitely) complete, and let $X$ be an object of $C$. Then the forgetful functor

$$\sum _X:C/X\to C$$\sum_X: C/X \to C

preserves and reflects (finite) connected limits.

Proof

The wide pullback of the diagram $f_i:X_i\to X$ where $f_i=1_X$ for all $i$ is clearly $X$, as is the equalizer of the pair of identity arrows. Since the product functor $C\times C\to C$ preserves arbitrary limits, we see that

$$X\times \lim (c_i\stackrel{g_i}{\to }c)=(\lim X_i\stackrel{f_i=1_X}{\to }X)\times (\lim c_i\stackrel{g_i}{\to }c)=\lim X\times c_i\stackrel{1\times g_i}{\to }X\times c,$$X \times lim (c_i \stackrel{g_i}{\to} c) = (lim X_i \stackrel{f_i = 1_X}{\to} X) \times (lim c_i \stackrel{g_i}{\to} c) = lim X \times c_i \stackrel{1 \times g_i}{\to} X \times c,

i.e., $X\times -$ preserves wide pullbacks. The same line of argument shows $X\times -$ preserves equalizers, so $X\times -$ preserves connected limits.

The functor $X\times -$ carries a canonical comonad structure, whose category of coalgebras is $C/X$. The forgetful functor $C/X\to C$ is comonadic, and thus preserves and reflects any class of limits preserved by the comonad $X\times -$. Thus ${\sum }_X:C/X\to C$ preserves and reflects all connected limits.

Preservation from wide pullbacks

Recall that a wide pullback is a limit over a diagram whose underlying shape is the poset obtained by freely adjoining a terminal element to a discrete poset, which is certainly connected.

It is not true that if $C$ has wide pullbacks then it has connected limits. The saturation of the class of wide pullbacks is the class of connected and “simply connected” limits (limits over categories $C$ whose groupoid reflection ${\Pi }_1(C)$ is trivial).

However, the following is true.

Theorem

Let $C$ be a complete category, and let $D$ be locally small. Then a functor $G:C\to D$ preserves connected limits if and only if it preserves wide pullbacks.

Proof

The forward direction is clear since wide pullbacks are examples of connected limits. Now suppose $G:C\to D$ preserves wide pullbacks. Then

(1)$$C\stackrel{G}{\to }D\stackrel{\mathrm{hom}(d,-)}{\to }\mathrm{Set}$$C \stackrel{G}{\to} D \stackrel{hom(d, -)}{\to} Set

preserves wide pullbacks for every object $d$ of $D$. Put $I=\mathrm{hom}(d,G1)$. The underlying functor

$$\sum :\mathrm{Set}/I\to \mathrm{Set}$$\sum\colon Set/I \to Set

reflects and preserves connected limits and in particular wide pullbacks, so that the evident lift

$$\mathrm{hom}(d,G-):C\to \mathrm{Set}/I$$\hom(d, G-)\colon C \to Set/I

preserves wide pullbacks. It also preserves the terminal object, hence by this proposition it preserves arbitrary limits. Therefore the composite

$$C\stackrel{\mathrm{hom}(d,G-)}{\to }\mathrm{Set}/I\stackrel{\sum }{\to }\mathrm{Set}$$C \stackrel{\hom(d, G-)}{\to} Set/I \stackrel{\sum}{\to} Set

preserves connected limits, for every object $d$. Since this is the same composite as in (1), and since the representables $\mathrm{hom}(d,-)$ jointly reflect arbitrary limits, we conclude that $G$ preserves connected limits.

In particular, for $C$ complete, a functor $G:C\to D$ that preserves wide pullbacks also preserves equalizers.

Related pages

• parametric right adjoint

References

• Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS