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April 3, 2017

Enrichment and its Limits

Posted by Emily Riehl

Guest post by David Jaz Myers

What are weighted limits and colimits? They’re great, that’s what!

In this post, we prepare for the next part of the Kan Extension Seminar by learning a bit about enrichment and weighted limits and colimits. I’ll also describe the “𝒱\mathcal{V}\, point of view” that I’ll be adopting for the next post.

Enriched Categories

Normally, we like to think that there is a set 𝒞(A,B)\mathcal{C}(A, B) of arrows between any two objects AA and BB in a category 𝒞\mathcal{C}. Composition then can be packaged up into a function 𝒞(A,B)×𝒞(B,C)𝒞(A,C)\mathcal{C}(A, B) \times \mathcal{C}(B, C) \to \mathcal{C}(A, C), satisfying an associativity condition. The identity is an element of 𝒞(A,A)\mathcal{C}(A, A), which we can associate with a function 1𝒞(A,A)1 \to \mathcal{C}(A, A) from the terminal object. Note that 11 acts as a unit (up to isomorphism) for the cartesian product ×\times.

We can abstract the structure of 11 and ×\times needed to express the axioms of a category as above and describe it in any category. We don’t need the limit properties of 11 and ×\times to describe a category, so we won’t keep them as part of our abstraction. What we are left with is the notion of a monoidal category, a category equipped with an object 11 (not necessarily terminal!) and a binary operation \otimes which satisfy the axioms of a monoid up to isomorphism (and then some conditions on those isomorphisms). We’ll ask for our monoidal categories to be symmetric, meaning that ABBAA \otimes B \cong B \otimes A (in a nice, natural way).

Expressing the axioms of a category in more general monoidal categories (𝒱,,1)(\mathcal{V}, \otimes, 1) gives us categories enriched in 𝒱\mathcal{V}. We call 𝒱\mathcal{V} the base category. Here are some examples of enriched categories over various bases to keep in mind:

  • (𝒱=Set\mathcal{V} = \text{Set}) If the base category is the category of sets, then our categories are the ones we are familiar with. In particular, all the other sorts of base categories are presumed to be enriched over the category of sets.
  • (𝒱=Ab\mathcal{V} = \text{Ab}) If the base category is abelian groups with their tensor product \otimes (and unit Z\Z), then we can add and subtract arrows ABA \to B in our categories. Furthermore, composition is bilinear with regards to the addition of arrows. An example of these sorts of categories are the categories of modules over a ring with the addition of arrows taken pointwise.
  • (𝒱=Top\mathcal{V} = \text{Top}) If our base category is topological spaces with its cartesian structure, then we have a full space of arrows between any two objects in our categories. Furthermore, composition is continuous.
  • (𝒱={falsetrue}\mathcal{V} = \{\false \to \true\}) If our base category is the truth values with “and” as their product and true\true as the unit, then our categories are just orders. The object of arrows 𝒞(A,B)\mathcal{C}(A, B) is a truth value, and we interpret it as the truth of the statement that AA is at most BB. For more on category theory over the truth values, check out Simon Willerton’s post on this blog!
  • (𝒱=[0,]\mathcal{V} = [0,\infty]) This example is rather surprising. If our base category is the non-negative real numbers, including infinity, with the ‘greater than’ ordering and ‘plus’ as its monoidal structure, then our categories are directed metric spaces. The object of arrows 𝒞(A,B)\mathcal{C}(A, B) is a non-negative real number which we interpret as the distance from AA to BB. William Lawvere was the first to write about this in his paper Metric Spaces, Generalized Logic, and Closed Categories

In order for the theory of enriched categories to work out nicely, we will ask for two further conditions on our base categories: closure and cocompleteness. A monoidal category 𝒱\mathcal{V} is closed if the functors A- \otimes A have right adjoints 𝒱(A,)\mathcal{V}(A, -). These objects 𝒱(A,B)\mathcal{V}(A, B) therefore behave as an internal hom, an object consisting of morphisms from AA to BB.

To see why, let’s briefly introduce the notation 𝒱 0(A,B)\mathcal{V}_0(A, B) to denote the set of morphisms from AA to BB in 𝒱\mathcal{V}. Note that 1AA1 \otimes A \cong A (by assumption), so 𝒱 0(1A,B)𝒱 0(A,B)\mathcal{V}_0(1 \otimes A, B) \cong \mathcal{V}_0(A, B). But by definition, then 𝒱 0(A,B)𝒱 0(1,𝒱(A,B))\mathcal{V}_0(A, B) \cong \mathcal{V}_0(1, \mathcal{V}(A, B)); in other words, the set of morphisms from AA to BB in 𝒱\mathcal{V} is the same as the set of points 1𝒱(A,B)1 \to \mathcal{V}(A, B).

Being cocomplete means that we can glue the objects of our base together. We use this all the time behind the scenes in category theory. Note that since A- \otimes A is a left adjoint, it commutes with colimits; this is like the distribution of multiplication over addition.

From now on, all our base categories will be assumed to be symmetric monoidal closed and cocomplete.

Remark Even though we started with the monoidal structure \otimes above, sometimes it is more natural to start with the internal hom and then define the monoidal structure to be its left adjoint. For example, it’s pretty easy to see how to endow the set of homomorphisms between two abelian groups with the structure of an abelian group: just add pointwise! Then, we can define the tensor product ABA \otimes B to be that abelian group which represents Ab(A,Ab(B,C))\text{Ab}(A, \text{Ab}(B, C)). This shows that maps out of ABA \otimes B must be bilinear; they are linear first in AA, and then in BB.

Weighted Limits and Colimits

Enriched category theory (over a general base 𝒱\mathcal{V}) plays out much like usual category theory over the sets, so long as everything is proved in a nicely categorical manner. One major difference, however, is the behavior of (co)limits.

In the category of sets, functions are determined by their actions on points. But in more general base categories, morphisms (by which I mean points of the set of morphisms) may be equal on points (by which I mean morphisms from the monoidal unit 11) but differ in other ways. For this reason, we need to update our notion of (co)limit for the enriched context by letting the cones have a fatter sort of shape. We call this enriched notion of (co)limit a weighted (co)limit. Kelly calls it an indexed (co)limit. (I would like to thank Pierre for the following discussion about defining cones in the enriched context)

The limit of a diagram D:𝒟𝒞D : \mathcal{D} \to \mathcal{C} is more correctly a limiting cone; the object that we call the limit sits atop a “wireframe” cone that projects down toward the diagram. Usually, we define a cone slickly as a natural transformation ΔCD\Delta C \to D from a constant diagram at an object CC in 𝒞\mathcal{C} to the diagram DD. The universal property of the limit then looks like this: 𝒞(C,limD)𝒞 𝒟(ΔC,D).\mathcal{C}(C, \text{lim} D) \cong \mathcal{C}^{\mathcal{D}}(\Delta C, D).

To define the constant diagram, we collapse all the arrows of 𝒟\mathcal{D} onto the identity of CC; this requires being able to “forget” the data in an object of our base. Arrow theoretically, we have 𝒟(i,j)1𝒞(C,C)\mathcal{D}(i, j) \to 1 \to \mathcal{C}(C,C) with the first arrow being the unique map guaranteed by the universal property of the terminal object. But in the enriched context, our monoidal unit 11 may not be terminal, so we won’t be able to do this in general! We need to rephrase our definition of the limit.

We can do this by working with diagrams in 𝒱\mathcal{V} (where we have access to the objects 𝒞(A,B)\mathcal{C}(A, B)), instead of of diagrams in 𝒞\mathcal{C}. 𝒞(C,limD)𝒱 𝒟(1,𝒞(C,D()).\mathcal{C}(C, \lim D) \cong \mathcal{V}^{\mathcal{D}}(1, \mathcal{C}(C, D(-)). This says that maps into the limit (in 𝒞(C,limD)\mathcal{C}(C, \lim D)) correspond to cones over the diagram (points of 𝒞(C,D())\mathcal{C}(C, D(-)) in the category of diagrams in the base). Saying that the cone has a “wireframe” shape just means that we are looking at points 1𝒞(C,D())1 \to \mathcal{C}(C, D(-)). For a given object ii in 𝒟\mathcal{D}, we then get a point 1𝒞(C,D i)1 \to \mathcal{C}(C, D_i), a single morphism extending from CC to D iD_i. But points are often not sensitive enough over a general base, so for a nicely enriched notion of limit, we will need to consider more general figures W i𝒞(C,D i)W_i \to \mathcal{C}(C, D_i). Here, W:𝒟𝒱W : \mathcal{D} \to \mathcal{V} is a functor of weights, and the universal property of the weighted limit is 𝒞(C,lim WD)𝒱 𝒟(W(),𝒞(C,D()).\mathcal{C}(C, \lim_W D) \cong \mathcal{V}^{\mathcal{D}}(W(-), \mathcal{C}(C, D(-)).

Weighted colimits are defined by the dual formula, with morphisms coming out of the diagram. Conical (“wireframe”) (co)limits are just weighted limits whose weights are constant at the monoidal identity.

Over a general base, we can consider a weighted limit of a particularly simple sort. Let 𝒟\mathcal{D} be the category with a single object and its identity arrow, so that a diagram D:𝒟𝒞D : \mathcal{D} \to \mathcal{C} is just an object of 𝒞\mathcal{C}. Then, given a weight W:𝒟𝒱W : \mathcal{D} \to \mathcal{V}, the universal property says 𝒞(C,D W)𝒱 𝒟(W,𝒞(C,D)),\mathcal{C}(C, D^W) \cong \mathcal{V}^{\mathcal{D}}(W, \mathcal{C}(C, D)), where I have taken the liberty of renaming the limit D WD^W because this is precisely the universal property of the power! If sets are our base, then D WD^W is a product of WW copies of the object DD, which justifies the name “power”.

Dually, the copower WDW \cdot D is the weighted colimit of the above data, and over sets it is the coproduct of WW copies of DD. But beware; we need the weights to be contravariant for a colimit, so that W()W(-) and 𝒞(D(),C)\mathcal{C}(D(-), C) have the same variance.

Taking weighted (co)limits is in fact functorial in the weights. For colimits this relationship is covariant; for limits it is contravariant. This means that for suitably cocomplete 𝒞\mathcal{C}, we get a functor 𝒱 𝒟 op𝒞\mathcal{V}^{\mathcal{D}^{\text{op}}} \to \mathcal{C} for any diagram D:𝒟𝒞D : \mathcal{D} \to \mathcal{C} which sends a weight to the colimit of DD weighted by it. This operation corresponds to left Kan extension of the diagram along the Yoneda embedding. Dually, right Kan extension can be expressed by taking weighted limits. As a corollary, we see that all concepts are weighted (co)limits.

An Example: Limits are Weighted Limits!

Here’s a very cool example of a weighted limit which finally lets us tie categorical limits to their analytic analogs. Recall that if our base is the non-negative real numbers [0,][0, \infty], ordered by \geq and with monoidal structure ++ and 00, then categories are directed metric spaces. A functor is then a map which does not increase distance between points. We’ll show that certain weighted limits of sequences are their limits in the analytic sense.

Consider the discrete metric space 𝒟\mathcal{D} whose points are the natural numbers and where 𝒟(n,m)=\mathcal{D}(n, m) = \infty for all nn and mm. A diagram D:𝒟𝒞D : \mathcal{D} \to \mathcal{C} is therefore simply a sequence in 𝒞\mathcal{C}; since the distance between any two points of 𝒟\mathcal{D} is infinite, no function can increase distance, so all functions are functors. Suppose we have a decreasing sequence W:𝒟[0,]W : \mathcal{D} \to [0,\infty] whose limit, in the analytic sense, is 00. The universal property of the limit of DD, weighted by WW, is then 𝒞(C,lim WD)[0,] 𝒟(W(),𝒞(C,D())),\mathcal{C}(C, \lim_W D) \cong [0,\infty]^{\mathcal{D}}(W(-), \mathcal{C}(C, D(-))), which, with a little jiggling of the abstract definitions into their specializations, becomes 𝒞(C,lim WD)=inf i𝒟(W i𝒞(C,D i)).\mathcal{C}(C, \lim_W D) = \text{inf}_{i \in \mathcal{D}}(W_i - \mathcal{C}(C, D_i)). In particular, 0=𝒞(lim WD,lim WD)=inf i𝒟(W i𝒞(lim WD,D i)),0 = \mathcal{C}(\lim_W D, \lim_W D) = \text{inf}_{i \in \mathcal{D}}(W_i - \mathcal{C}(\lim_W D, D_i)), which shows that the sequence DD approaches lim WD\lim_W D in the analytic sense! (That subtraction is the internal hom in [0,][0, \infty]. It is truncated, so if it were going to be negative it gets set to 00.) It can, in fact, be shown that a sequence is Cauchy if and only if such a weight exists; see this very cool paper by Rutten for details.

If you enjoyed this example, check out Simon Willerton’s posts about the Legendre-Fenchel transform on this very blog!

The 𝒱\mathcal{V} Point of View

As a general motto, when we are working over a base category 𝒱\mathcal{V}, the category 𝒱\mathcal{V}, as a category enriched over itself, “thinks” it is the category of sets. By this I mean that if we work only using the tensor and internal hom of 𝒱\mathcal{V} to discuss things, then many of the peculiar features that the category of sets has as a (set-)category, the category 𝒱\mathcal{V} has as a 𝒱\mathcal{V}-category. For example:

  • As I mentioned above, functions in the category of sets are determined by their actions on points. This is not true of a general base category if we think of it as a category over sets. But, if we think of 𝒱\mathcal{V} as a category over itself, then we have that 𝒱(1,X)X\mathcal{V}(1, X) \cong X naturally in XX. By 𝒱(1,X)\mathcal{V}(1, X) here I mean the internal hom of 𝒱\mathcal{V}. This means that from the 𝒱\mathcal{V}-point of view, a morphism is determined by its action on points.
  • We can build on the last bullet point. We like to think of sets as totally discrete; they are, in fact, all disjoint unions of points. This totally fails in a general base category if we think of it as a category over the sets; for example, not all abelian groups are free (that is, not all abelian groups are coproducts of the monoidal unit Z\Z). But, if we work over a base category 𝒱\mathcal{V}, then “disjoint union” should really mean “copower”. In 𝒱\mathcal{V}, the copower is just the tensor product and therefore V1VV \cdot 1 \cong V. So, from the 𝒱\mathcal{V}-point of view, every object is a sum of points.

Check out the next post to see some of the cool things you can do from the 𝒱\mathcal{V} point of view!

Posted at April 3, 2017 1:16 AM UTC

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Re: Enrichment and its Limits

To define the constant diagram, we collapse all the arrows of 𝒟\mathcal{D} onto the identity of A

(and ff.) should be identity of C, right?

this was a really nice intro though, thanks. I look forward now to learning about gluing shapes.

Posted by: Matt Earnshaw on April 3, 2017 10:09 PM | Permalink | Reply to this

Re: Enrichment and its Limits

Yes, it should be! Thanks, I’ll correct that.

And thank you, hope you enjoy the shapes.

Posted by: David Jaz Myers on April 3, 2017 10:27 PM | Permalink | Reply to this

Re: Enrichment and its Limits

Hi David, great post! Do you know if this can be done in the context of enrichment over monoidal (,1)(\infty,1)-categories?

Posted by: Jacob A Gross on April 4, 2017 5:14 AM | Permalink | Reply to this

Re: Enrichment and its Limits

Thanks for this prequel to your main post. If we think of colimits as generalizations of sums, then weighted colimits ought to be generalizations of weighted sums iw iv i\sum_i w_i \mathbf{v}_i. The coend formula for weighted colimits gives weight to this analogy:

(1)colim WF d𝒟W(d)F(d). \text{colim}^W F \cong \int^{d \in \mathcal{D}} W (d) \cdot F (d).

Here, W:𝒟 op𝒱W: \mathcal{D}^{op} \to \mathcal{V} specifies the coefficients or weights W(d)𝒱W (d) \in \mathcal{V}, while F:𝒟𝒞F: \mathcal{D} \to \mathcal{C} specifies the ‘vectors’ F(d)𝒞F (d) \in \mathcal{C} that we are taking a weighted sum of. (For the formula to make sense, we need to assume that 𝒞\mathcal{C} is cocomplete and tensored over 𝒱\mathcal{V}, so let’s suppose this is true for the rest of this comment. I’ll also assume that 𝒱\mathcal{V} is a cosmos.)

There’s more to this analogy, which I first saw in a MathSE answer by Alexander. In hindsight, it’s clear from the way things work out that this analogy must have been the guiding principle behind presheaf categories, the Yoneda lemma, Yoneda reduction, (co)ends, weighted (co)limits and Kan extensions. But the textbooks don’t say so! So I thought it might be worth working out some of the details, and in particular to elaborate on the paragraph:

This means that for suitably cocomplete 𝒞\mathcal{C}, we get a functor 𝒱 𝒟 op𝒞\mathcal{V}^{\mathcal{D}^{\text{op}}} \to \mathcal{C} for any diagram D:𝒟𝒞D : \mathcal{D} \to \mathcal{C} which sends a weight to the colimit of DD weighted by it. This operation corresponds to left Kan extension of the diagram along the Yoneda embedding. … As a corollary, we see that all concepts are weighted (co)limits.

I’ll do this using coends, weighted colimits and left Kan extensions, as they seem to fit this analogy better than ends, weighted limits and right Kan extensions.

Given a 𝒱\mathcal{V}-category 𝒟\mathcal{D}, we can form the presheaf category [𝒟 op,𝒱]=𝒱 𝒟 op[\mathcal{D}^{op}, \mathcal{V}] = \mathcal{V}^{\mathcal{D}^{op}}. The Yoneda embedding Y:d𝒟(,d)Y: d \mapsto \mathcal{D}(-,d) allows us to think of 𝒟\mathcal{D} as sitting inside [𝒟 op,𝒱][\mathcal{D}^{op}, \mathcal{V}] as the representable functors. What’s nice about [𝒟 op,𝒱][\mathcal{D}^{op}, \mathcal{V}] is that we can take take sums - and more generally, colimits - of things in 𝒟\mathcal{D}. In fact, every W[𝒟 op,𝒱]W \in [\mathcal{D}^{op}, \mathcal{V}] can be expressed as a weighted colimit of these representables,

(2)Wcolim WY dW(d)𝒟(,d) W \cong \text{colim}^W Y \cong \int^{d} W (d) \cdot \mathcal{D}(-, d)

so we can think of the representables 𝒟(,d)\mathcal{D}(-,d) as being a basis of [𝒟 op,𝒱][\mathcal{D}^{op}, \mathcal{V}]. In some sense, the coend formula for the weighted colimit in (2) is the only one we need to know: other formulas such as (1) follow from it by ‘extending linearly’!

For example, given another functor F:𝒟𝒞F: \mathcal{D} \to \mathcal{C}, it makes sense to ask if we can ‘linearly extend’ FF to get a functor F^:[𝒟 op,𝒱]𝒞\hat{F}: [\mathcal{D}^{op}, \mathcal{V}] \to \mathcal{C}. Such an F^\hat{F} ought to agree with FF on the basis elements, so we want

(3)F^(𝒟(,d))=F(d). \hat{F} ( \mathcal{D}(-,d) ) = F (d).

Then, given some other W[𝒟 op,𝒱]W \in [\mathcal{D}^{op}, \mathcal{V}], we can just ‘extend linearly’, so that

(4)F^(W)=F^( dW(d)𝒟(,d))= dW(d)F^(𝒟(,d))= dW(d)F(d) \hat{F} (W) = \hat{F} \left(\int^{d} W (d) \cdot \mathcal{D}(-, d) \right) = \int^d W (d) \cdot \hat{F} ( \mathcal{D}(-, d) ) = \int^{d} W (d) \cdot F (d)

But this is just colim WF\text{colim}^W F in (1)! And as David says in his post, F^\hat{F} is the left Kan extension of the diagram F:𝒟𝒞F: \mathcal{D} \to \mathcal{C} along the Yoneda embedding:

(5)F^=Lan YF, \hat{F} = \text{Lan}_Y F,

so left Kan extensions are the analogue of extending linearly! Here’s a summary diagram:

Weighted Colimits

Not only is this an example of a left Kan extension, it’s the most important one: the coend formula for other Kan extensions factors through it! Given p:𝒟𝒟p: \mathcal{D} \to \mathcal{D}', the left Kan extension of F:𝒟𝒞F: \mathcal{D} \to \mathcal{C} along pp is given by

(6)Lan pF(d) d𝒟𝒟(p(d),d)F(d). \text{Lan}_p F (d') \cong \int^{d \in \mathcal{D}} \mathcal{D}' (p(d), d') \cdot F(d).

Let’s derive this formula. The functor p:𝒟𝒟p: \mathcal{D} \to \mathcal{D}' induces a functor p *:[𝒟 op,𝒱][𝒟 op,𝒱]p^*: [\mathcal{D}'\,^{op}, \mathcal{V}] \to [\mathcal{D}^{op}, \mathcal{V}] given by precomposing with pp. Composing this functor with the Yoneda embedding Y Y' of 𝒟\mathcal{D}', we get a functor 𝒟[𝒟 op,𝒱]\mathcal{D}' \to [\mathcal{D}^{op}, \mathcal{V}]:

(7)d𝒟(,d)p=𝒟(p(),d). d' \mapsto \mathcal{D}'(-, d') \circ p = \mathcal{D}'(p(-), d').

Evaluating Lan YF\text{Lan}_Y F at this weight yields the formula in (6)! In summary, we have:

Left Kan Extension

The bottom left triangle doesn’t commute on the nose, but we have an obvious natural transformation Yp *YpY \Rightarrow p^* \circ Y' \circ p whose components 𝒟(c,d)𝒟(p(c),p(d))\mathcal{D}(c,d) \to \mathcal{D}'(p(c), p(d)) are given by pp. This induces the natural transformation η:FLan pFp\eta: F \Rightarrow \text{Lan}_p F \circ p of the left Kan extension.

The way I’ve written it, it seems like all Kan extensions Lan pF\text{Lan}_p F exist as long as Lan YF\text{Lan}_Y F exists, but that doesn’t sound right. Maybe someone else can highlight all the ‘existential’ nuances that I’ve missed.

Posted by: Ze on April 5, 2017 2:32 AM | Permalink | Reply to this

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