nLab
connected limit

Contents

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:

Properties

Construction from pullbacks and equalizers

Theorem

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

Proof

Let II be a finite connected category and F:ICF\colon I\to C a functor. Since II is connected, it is inhabited; choose some object x 0Ix_0\in I. For each object xIx\in I, let (x)\ell(x) be the length of the shortest zigzag from x 0x_0 to xx. Now order the objects of II as x 0,x 1,,x nx_0, x_1, \dots, x_n such that for all ii we have (x i)(x i+i)\ell(x_i) \le \ell(x_{i+i}).

Now we inductively define objects P iCP_i\in C, for 0in0\le i\le n, with projections p ij:P iF(x j)p_{i j}\colon P_i \to F(x_j) for jij\le i. We begin with P 0=x 0P_0 = x_0 and p 00=1 x 0p_{0 0}= 1_{x_0}. Assuming P iP_i and p ijp_{i j} defined, choose a zigzag from x 0x_0 to x i+1x_{i+1} of minimal length, say

x 0y 1y kx 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 II, we have y k=x jy_k = x_j for some jij\le i, and thus we have q=p ij:P iy kq = p_{i j}\colon P_i \to y_k.

If the final arrow y kx i+1y_k \leftrightarrow x_{i+1} in the zigzag is directed as y kx i+1y_k \to x_{i+1}, then let P i+1=P iP_{i+1} = P_i, let p i+1,i+1p_{i+1, i+1} be the composite P iqF(y k)F(x i+1)P_i \xrightarrow{q} F(y_k) \to F(x_{i+1}), and keep the other p ijp_{i j} unchanged. On the other hand, if y kx i+1y_k \leftrightarrow x_{i+1} is directed as y kx i+1y_k \leftarrow x_{i+1}, let P i+1P_{i+1} be the pullback

P i+1 P i p i+1,i+1 q F(x i+1) F(y k) \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,jp_{i+1,j} for jij\le i by composition with P i+1P iP_{i+1}\to P_i.

At the end of this procedure, we have an object P nP_n with projections p n,j:P nF(x j)p_{n,j}\colon P_n \to F(x_j) for all objects x jIx_j\in I. Now we order the morphisms in II as g 0,,g mg_0,\dots,g_m and define, inductively, an object Q iQ_i with a morphism q i:Q iP nq_i\colon Q_i \to P_n. We begin with Q 0=P nQ_0 = P_n and q 0=1 P nq_0 = 1_{P_n}. Given Q iQ_i and q iq_i, we let e:Q i+1Q ie\colon Q_{i+1} \to Q_{i} be the equalizer of F(g i+1)p n,?q iF(g_{i+1}) \circ p_{n,?} \circ q_i and p n,?q ip_{n,?} \circ q_i, where the ?s denote the indices of the objects that are the source and target of g i+1g_{i+1}. We then set q i+1=q ieq_{i+1} = q_i \circ e.

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

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

Theorem

Let CC be (finitely) complete, and let XX be an object of CC. Then the forgetful functor

X:C/XC\sum_X: C/X \to C

preserves and reflects (finite) connected limits.

Proof

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

X×lim(c ig ic)=(limX if i=1 XX)×(limc ig ic)=limX×c i1×g iX×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×X \times - preserves wide pullbacks. The same line of argument shows X×X \times - preserves equalizers, so X×X \times - preserves connected limits.

The functor X×X \times - carries a canonical comonad structure, whose category of coalgebras is C/XC/X. The forgetful functor C/XCC/X \to C is comonadic, and thus preserves and reflects any class of limits preserved by the comonad X×X \times -. Thus X:C/XC\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 CC 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 CC whose groupoid reflection Π 1(C)\Pi_1(C) is trivial).

However, the following is true.

Theorem

Let CC be a complete category, and let DD be locally small. Then a functor G:CDG\colon 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:CDG\colon C \to D preserves wide pullbacks. Then

(1)CGDhom(d,)SetC \stackrel{G}{\to} D \stackrel{hom(d, -)}{\to} Set

preserves wide pullbacks for every object dd of DD. Put I=hom(d,G1)I = hom(d, G 1). The underlying functor

:Set/ISet\sum\colon Set/I \to Set

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

hom(d,G):CSet/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

Chom(d,G)Set/ISetC \stackrel{\hom(d, G-)}{\to} Set/I \stackrel{\sum}{\to} Set

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

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

References

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

Revised on February 16, 2013 19:32:13 by Todd Trimble (98.208.182.196)