nLab
weighted limit

Idea

The notion of weighted limit (also called indexed limit) is naturally understood from the point of view on limits as described at representable functor:

Weighted limits make sense and are considered in the general context of $V$-enriched category theory, but restrict attention to $V=$ Set for the moment, in order to motivate the concept.

Let $K$ denote the small category which indexes diagrams over which we want to consider limits and eventually weighted limits. Notice that for

$$F \colon K \to Set$$

a Set-valued functor on $K$, the limit of $F$ is canonically identified simply with the set of cones with tip the singleton set $pt = \{\bullet\}$:

$$lim F = [K,Set](\Delta pt, F) \,.$$

This means, more generally, that for

$$F \colon K \to C$$

a functor with values in an arbitrary category $C$, the object-wise limit of the functor $F$ under the Yoneda embedding

$$C(-,F(-)) : K \stackrel{F}{\to} C \stackrel{Y}{\to} Set^{C^{op}}$$

which appears in the discussion in this example at representable functor can be expressed by the right side of

$$lim C(-,F(-)) = [K,Set](\Delta pt, C(-,F(-))) \,.$$

(Recall that this is the limit over the diagram $C(-,F(-)) \colon K \to Set^{C^{op}}$ which, if representable defines the desired limit of $F$.)

The idea of weighted limits is to

1. allow in the formula above the particular functor $\Delta pt$ to be replaced by any other functor $W \colon K \to Set$;

2. to generalize everything straightforwardly from the Set-enriched context to arbitrary $V$-enriched contexts (see below).

The idea is that the weight $W \colon K \to V$ encodes the way in which one generalizes the concept of a cone over a diagram $F$ (that is, something with just a tip from which morphisms are emanating down to $F$) to a more intricate structure over the diagram $F$. For instance in the application to homotopy limits discussed below with $V$ set to SimpSet the weight is such that it ensures that not only 1-morphisms are emanating from the tip, but that any triangle formed by these is filled by a 2-cell, every tetrahedron by a 3-cell, etc.

Definition

Let $V$ be a closed symmetric monoidal category. All categories in the following are $V$-enriched categories, all functors are $V$-functors.

A weighted limit over a functor

$$F : K \to C$$

with respect to a weight or indexing type functor

$$W : K \to V$$

is, if it exists, the object $lim^W F \in C$ which represents the functor (in $c \in C$)

$$[K,V](W, C(c,F(-))) : C^{op} \to V \,,$$

i.e. such that for all objects $c \in C$ there is an isomorphism

$$C(c, lim^W F) \simeq [K,V](W(-), C(c,F(-)))$$

natural in $c$.

(Here $[K,V]$ is the $V$-enriched functor category, as usual.)

In particular, if $C = V$ itself, then we get the direct formula

$$lim^W F \simeq [K,V](W,F) \,.$$

This follows from the above by the end manipulation

$$\begin{aligned} [K,V](W(-),C(c,F(-))) &:= \int_{k \in K} V(W(k),V(c,F(k))) \\ & \simeq \int_{k \in K} V(c,V(W(k),F(k)) \\ & \simeq V(c, \int_{k \in K} V(W(k),F(k)) \\ & =: V(c, [K,V](W,F)) \,. \end{aligned}$$

Motivation from enriched category theory

Let $V$ be a monoidal category.

Imagine you’re tasked to write down the definition of limit in a category $C$ enriched over $V$. You would start saying there is a diagram $F \colon K \to C$ and a limit is a universal cone over it, i.e. it’s the universal choice of an object $c$ together with an arrow $f_k \colon c \to F(k)$ for each object $k$ of $K$.

Here’s where you stop and ask yourself: what is ‘an arrow’ in $C$? $C$ has no hom-sets — it has hom-objects — hence what’s ‘an element’ of $C(c, F(k))$ in $V$?

There are two ways to specify an element of an object $X$ in a monoidal category $(V, I, \otimes)$:

1. Give an arrow $I \to X$ (think of sets, where elements of $X$ are indeed the same thing as arrows $\{*\} \to X$. These are called global elements of $X$, and are more often than not a misbehaved notion of element, since often $I$ is ‘too big’ to thoroughly probe $X$ (on the other hand, notice the underlying category of an enriched category is defined by taking global elements of the hom-objects)
2. Give any arrow into $X$. These are called generalized elements, and the existence of the Yoneda embedding assures us they completely capture the categorical structure of $V$.

Hence you now say: a cone over $F$ is a choice of a generalized element $f_k$ of $C(c, F(k))$, for every $k$ in $K$. This means specifying an arrow $W_k \to C(c, F(k))$ in $V$, for each $k$. It’s now quite natural to ask for the functoriality of this choice in $k$, hence we end up defining a ‘generalized cone’ over $F$ as an element

$$[K, V](W(-), C(c, F(-)))$$

Hence $W$ is simply a uniform way to specify the sides of a cone. A confirmation that this is indeed the right definition of limit in the enriched settings come from the fact that ‘conical completeness’ (a conical limit now is one where $W = \Delta I$, hence we pick only global element) is an inadequate notion, see for example Section 3.9 in Kelly’s book (aptly named The inadequacy of conical limits).

Examples

Homotopy limits

For $V$ some category of higher structures, the local definition of homotopy limit over a diagram $F : K \to C$ replaces the ordinary notion of cone over $F$ by a higher cone in which all triangles of 1-morphisms are filled by 2-cells, all tetrahedra by 3-cells, etc.

One can convince oneself that for the choice of SimpSet for $V$ this is realized in terms of the weighted limit $lim^W F$ with the weight $W$ taken to be

$$W : K \to \Simp\Set$$

$$W : k \mapsto N(K/k) \,,$$

where $K/k$ denotes the over category of $K$ over $k$ and $N(K/k)$ denotes its nerve.

This leads to the classical definition of homotopy limits in $\Simp\Set$-enriched categories due to

• A.K. Bousfield and D.M. Kan, Homotopy limits, completions, and localizations

See for instance also

• Nicola Gambino, Weighted limits in simplicial homotopy theory (pdf or pdf)

In some nice cases the weight $N(K/-)$ can be replaced by a simpler weight; an example is discussed at Bousfield-Kan map.

Homotopy pullback

For instance in the case that $K = \{r \to t \leftarrow s\}$ is the shape of pullback diagrams we have

$$W(r) = \{r\}$$

$$W(s) = \{s\}$$

$$W(t) = N( \{r \to t \leftarrow s\} )$$

and $W(r \to t) : \{r\} \to \{r \to t \leftarrow s\}$ injects the vertex $r$ into $\{r \to t \leftarrow s\}$ and similarly for $W(s \to t)$.

This implies that for $F : K \to C$ a pullback diagram in the SimpSet-enriched category $C$, a $W$-weighted cone over $F$ with tip some object $c \in C$, i.e. a natural transformation

$$W \Rightarrow C(c, F(-))$$

is

• over $r$ a “morphism” from the tip $c$ to $F(r)$ (i.e. a vertex in the Hom-simplicial set $C(c,F(r))$);

• similarly over $s$;

• over $t$ three “morphisms” from $c$ to $F(t)$ together with 2-cells between them (i.e. a 2-horn in the Hom-simplicial set $C(c,F(t))$)

• such that the two outer morphisms over $t$ are identified with the morphisms over $r$ and $s$, respectively, postcomposed with the morphisms $F(r \to t)$ and $F(s \to t)$, respectively.

So in total such a $W$-weighted cone looks like

$$\array{ &&& c \\ & \swarrow &\Rightarrow& \downarrow &\Leftarrow& \searrow \\ F(r) && \stackrel{F(r \to t)}{\to} & F(t) & \stackrel{F(s \to t)}{\leftarrow} && F(s) }$$

as one would expect for a “homotopy cone”.

References for homotopy limits in terms of weighted limits

Details of this are discussed for instance in the book

• Hirschhorn, Model categories and their localization

To compare with the above discussion notice that

• The functor

  $$W := N(K/-)$$

  is discussed there in definition 14.7.8 on p. 269.

• the $V$-enriched hom-category $[K,V]$ which on $V$-functors $S,T$ is the end $[K,V](S,T) = \int_{k \in K} V(S(k), T(k))$ appears as $hom^K(S,T)$ in definition 18.3.1 (see bottom of the page).

• for $V$ set to SimpSet the above definition of homotopy limit appears in example 18.3.6 (2).

Related pages

• strict 2-limit

• saturated class of limits

• weighted colimit

References

A standard reference is

• Max Kelly, section 3.1, p. 37 in: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp.; remake: TAC reprints 10 (tac:tr10, pdf)

In

• Emily Riehl, Weighted limits and colimits (2008) (pdf)

is given an account of lectures by Mike Shulman on the subject. The definition appears there as definition 3.1, p. 4 (in a form a bit more general than the one above).

The analogous notion of weighted (infinity,1)-limit is discussed in

• Martina Rovelli, Weighted limits in an (∞,1)-category, 2019, arxiv:1902.00805