nLab weighted limit

Idea

The notion of weighted limit (also called indexed limit or mean cotensor product in older texts) 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

$$C\big(-,F(-)\big) \,\colon\, K \overset{F}{\longrightarrow} C \overset{Y}{\longrightarrow} Set^{C^{op}}$$

can be expressed by the right hand side of:

(1)$$\underset{ \underset{k \in K}{\longleftarrow} }{lim} C\big(-,F(k)\big) \;=\; [K,Set]\Big( \Delta pt ,\, C\big(-,F(-)\big) \Big) \,.$$

(This is the limit over the diagram $C\big(-,F(-)\big) \,\colon\, K \to Set^{C^{op}}$ which, if representable, defines the desired limit of $F$, see this example at representable functor).

The idea of weighted limits is to:

1. allow in the formula (1) 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 cones 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 $V$ being 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 $V$-enriched functors.

A weighted limit over a functor

$$F \,\colon\, K \to C$$

with respect to a weight or indexing type functor

$$W \,\colon\, K \to V$$

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

$$[K,V]\Big(W, C\big(c,F(-)\big)\Big) \;\colon\; C^{op} \to V \,,$$

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

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

natural in $c$.

(Here $[K,V]$ denotes 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

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

Weighted limits for $V = Set$

Let us spell out what a weighted limit looks like in ordinary category theory, to give intuition for the difference between weighted limits and ordinary limits.

Given a weight $W : K \to Set$ and a diagram $F : K \to C$, a weighted limit comprises an object $L$ together with a projection $\pi_{k, w} : L \to F(k)$ for each $k \in K$ and $w \in W(k)$ such that the following diagram commutes for $k, k' \in K$, $w \in W(k)$ and $\kappa : k \to k'$: This is required to be universal in the sense that given every such diagram as above with domain $C$, there is a unique morphism $C \to L$ making the diagrams commute.

It is clear that when $W$ is the constant functor sending everything to a singleton set, this recovers the usual notion of limit for $F$.

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 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

• Jean-Marc Cordier and Timothy Porter, Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, (pdf)

• 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”.

Related pages

• strict 2-limit

• saturated class of limits

• weighted colimit

References

General

The notion of weighted limits was introduced (under the name “mean cotensor product”) in:

• Francis Borceux, p. 10 of: Une notion de $V$-limite, in Colloque sur l’algèbre des catégories. Amiens 1973. Résumés des conférences, Cahiers de topologie et géométrie différentielle 14 2 (1973) 153-223 [numdam:CTGDC_1973__14_2_153_0]

• Francis Borceux, Gregory Maxwell Kelly, A notion of limit for enriched categories, Bulletin of the Australian Mathematical Society 12 1 (1975) 49-72 [doi:10.1017/S0004972700023637]

and, independently, (under the name “Hom (formel)”) by:

• C. Auderset, Adjonction et monade au niveau des 2-categories, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20. (numdam)

Textbook accounts:

• 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), Reprints in Theory and Applications of Categories, 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]

• Francis Borceux, §6.6 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]

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).

Presenting homotopy limits

On weighted limits as presentations of homotopy limits:

• Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf, pdf]

To compare with the above discussion notice that

• The functor

  $$W \;\coloneqq\; N(K/-)$$

  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).

• Emily Riehl, §6.6 and Chapter 7 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

Discussion of weighted $(\infty,1)$-limits:

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