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March 24, 2014

An Exegesis of Yoneda Structures

Posted by Emily Riehl

Guest post by Alexander Campbell

We want to develop category theory in a general 2-category, in order to both generalise and clarify our understanding of category theory. The key to this endeavour is to express the basic notions of the theory of categories in a natural 2-categorical language. In this way we are continuing a theme present in previous posts from the Kan Extension Seminar, wherein monads and adjunctions were given a 2-categorical setting, and by analogy, in our very first paper, whose purpose was to express basic notions of the theory of sets in a natural categorical language. In this post we consider a concept very central and special to category theory: the Yoneda lemma.

So what’s the Yoneda Lemma again?

The Yoneda lemma says that for any object aa of a category AA, the diagram 1 a A * ι A(a,) Set \begin{matrix} 1 & \overset{a}{\rightarrow} & A \\ {}_{\ast} \searrow & \overset{\iota}{\Rightarrow} & \swarrow_{A(a,-)} \\ & Set \\ \end{matrix} is a left extension.

In this post I will give a motivation for the notion of Yoneda structure, as defined in the paper Yoneda Structures on 2-Categories of Ross Street and Bob Walters. But before we begin I would like to take this opportunity to thank Emily for inviting me to join the Kan Extension Seminar and for her support and encouragement throughout the course. This has been and continues to be a singularly valuable experience in my first year as a category theorist.

Liftings and extensions

After the various compositions available in a 2-category (nicely unified as the operation of pasting), the most basic notions of two-dimensional algebra are liftings and extensions. A diagram A h α g C f B \begin{matrix} & A \\ {}^h \swarrow & \overset{\alpha}{\Leftarrow} & \searrow^{g} \\ C & \overset{f}{\rightarrow} & B \\ \end{matrix} in a 2-category 𝒦\mathcal{K} is said to be a left lifting if pasting along hh effects a bijection of 2-cells hkgfk. \frac{h \Rightarrow k}{g \Rightarrow f k}.

There is a similar elementary definition of left extension, but we can simply say that a left extension is a left lifting in 𝒦 op\mathcal{K}^{op} (reverse 1-cells). Hence these are just dual versions of a single notion. Left liftings in 𝒦 co\mathcal{K}^{co} and 𝒦 coop\mathcal{K}^{coop} are right liftings and right extensions respectively, but these will not concern us here.

An elementary result of 2-categorical algebra is the pasting lemma. This tells us that given a diagram A h β g α f D l C k B \begin{matrix} & & A \\ {}^h \swarrow & \overset{\beta}{\Leftarrow} & \downarrow^{g} & \overset{\alpha}{\Leftarrow} & \searrow^{f} \\ D & \underset{l}{\rightarrow} & C & \underset{k}{\rightarrow} & B \\ \end{matrix} in which α\alpha is a left lifting, β\beta is a left lifting if and only if the composite is a left lifting. By duality, we immediately get a corresponding result for left extensions.

The Yoneda lemma and Yoneda structures

A Yoneda structure on a 2-category consists of two pieces of data satisfying three axioms. We will see that the data is what is necessary to naturally express the Yoneda lemma in the 2-category, and that the axioms are expressions of familiar results of category theory.

In a general 2-category, following the philosophy of generalised elements, it is natural to consider arrows XAX \longrightarrow A with arbitrary domain (not just restricted to the terminal object 1) as objects of AA. In order to state the Yoneda lemma for such generalised objects, we must replace *:1Set\ast \colon 1 \longrightarrow \text{Set} with some arrow y X:XPXy_X \colon X \longrightarrow P X. Here PXP X could be loosely thought of as “sets freely varying over XX”; in CAT, it is the category of presheaves over XX and y Xy_X is the Yoneda embedding.

But now we must address the issue of size. Firstly, in CAT, the Yoneda embedding y Xy_X exists only for locally small XX. Secondly, in the wish for a snappy statement of the Yoneda lemma, I left an important condition unmentioned: it is not true for a general category AA that the hom-functor A(a,)A(a,-) exists for every object aa. (It would be true if AA were locally small, but we do not want CAT to consist only of locally small categories; for if XX is not small, then PXP X is not locally small.) Hence we must restrict those elements for which we state the Yoneda lemma.

We can now specify the data of a Yoneda structure on a 2-category. The first piece of data is a class of admissible arrows in the 2-category; the only property we require of this class is that it be a right ideal, meaning that if gg is admissible, then so is gfg f for every such composable ff. We say that an object AA is admissible if the identity arrow 1 A:AA1_A \colon A \longrightarrow A is admissible; in the Yoneda structure on CAT, the admissible objects are the locally small categories. Hence we see that there is nothing mysterious about size; it is just part of the structure. The other piece of a Yoneda structure is the Yoneda arrows: an admissible arrow y A:APAy_A \colon A \longrightarrow P A for each admissible object AA.

With this structure we can naturally state the Yoneda lemma in a 2-category. We take this as our first axiom for a Yoneda structure.

Axiom 1. For XX, aa both admissible, the left extension X a A y X χ a A(a,1) PX \begin{matrix} X & \overset{a}{\rightarrow} & A \\ {}_{y_X} \searrow & \overset{\chi^a}{\Rightarrow} & \swarrow_{A(a,1)} \\ & P X \\ \end{matrix} exists.

The universal property of this extension is indicated by A(a,1)fy Xfa. \frac{A(a,1) \Rightarrow f}{y_X \Rightarrow f a}.

So we have used the Yoneda lemma as a definition of the hom-functors A(a,1)A(a,1); the axiom asserts their existence. We can further define the hom-functors on 1-cells and 2-cells in both variables: in the first variable by the universal property of extension, and in the second variable by composition. In particular, for (suitably admissible) f:ABf \colon A \longrightarrow B, we can define Pf:PBPAP f \colon P B \longrightarrow P A as Pf=(PB)(B(1,f),1)P f = (P B)(B(1,f),1), with a similar definition on 2-cells. This all just follows from Axiom 1, hence is part of any Yoneda structure.

All further use of the admissibility notion is just to ensure the existence of the data of Axiom 1. So I will make minimal explicit mention of it from here; it can be easily filled in as necessary.

Universal arrows and absolute liftings

The second axiom of a Yoneda structure is expressed in terms of liftings. I will now exhibit the presence of liftings in category theory.

The first formalisation of the notion of universal property that Mac Lane gives in Categories for the Working Mathematician is that of universal arrow (I will deal with the ‘left’ sense here). Given a functor f:ABf \colon A \longrightarrow B and an object bb of BB, he defines what it means for a “pair” consisting of an object aa of AA and an arrow θ:bfa\theta \colon b \longrightarrow fa to be a universal arrow from bb to ff. This definition says precisely that 1 a θ b A f B \begin{matrix} & 1 \\ {}^a \swarrow & \overset{\theta}{\Leftarrow} & \searrow^{b} \\ A & \underset{f}{\rightarrow} & B \\ \end{matrix} is a left lifting diagram.

In a general 2-category, the restriction of our consideration to such ‘global objects’ is untenable. So how do we express the universal property of a ‘generalised arrow’ from a ‘generalised object’ b:XBb \colon X \longrightarrow B to ff? It is not enough to say that the corresponding diagram X a θ b A f B \begin{matrix} & X \\ {}^a \swarrow & \overset{\theta}{\Leftarrow} & \searrow^{b} \\ A & \underset{f}{\rightarrow} & B \\ \end{matrix} is a left lifting diagram; we must consider the object bb at all its stages of development. Hence we define a 2-cell θ\theta to be a universal arrow from bb to ff when for every earlier stage YXY \longrightarrow X, the diagram Y X a θ b A f B \begin{matrix} & Y \\ & \downarrow \\ & X \\ {}^a \swarrow & \overset{\theta}{\Leftarrow} & \searrow^{b} \\ A & \underset{f}{\rightarrow} & B \\ \end{matrix} is a left lifting. That is, we require our corresponding diagram to be an absolute left lifting (meaning that the property of being a left lifting is preserved when composed with any arrow into XX).

Hence absolute liftings give a natural 2-categorical expression of the notion of universal arrow. This slogan can help us to interpret some basic facts of the algebra of 2-categories as they relate to category theory. For instance, the result that a 2-cell A f η 1 B g A \begin{matrix} & A \\ {}^f \swarrow & \overset{\eta}{\Leftarrow} & \searrow^{1} \\ B & \underset{g}{\rightarrow} & A \\ \end{matrix} is the unit of an adjunction fgf \dashv g if and only if it is an absolute left lifting corresponds to the familiar result of basic category theory that states that a functor g:BAg \colon B \longrightarrow A is a right adjoint if and only if there is a universal arrow to gg from every object of AA.

Universal elements and representability

Now, observe that the 2-cell in the ordinary Yoneda lemma, in addition to being a left extension, is trivially a left lifting; for this just means that there is a bijection between arrows aba \longrightarrow b and elements of the set A(a,b)A(a,b). In familiar terms, this says that ι\iota is a universal element of the functor A(a,)A(a,-), and by the above discussion we can immediately incorporate this into our setting as the second axiom for a Yoneda structure.

Axiom 2. The 2-cell χ a\chi^a of Axiom 1 is an absolute left lifting.

The universal property of this absolute left lifting is indicated by axbX(1,x)A(a,b). \frac{a x \Rightarrow b}{X(1,x) \Rightarrow A(a,b)}.

The combination of these two universal properties of the 2-cells χ a\chi^a is the ignition that allows us to begin to develop category theory in our 2-category. Together they give a bijection of 2-cells π\pi and η\eta as indicated by the bijections B(s,1)C(j,t)y AC(j,ts)jts \frac{\frac{B(s,1) \Rightarrow C(j,t)}{y_A \Rightarrow C(j,t s)}}{j \Rightarrow t s} and as displayed in the diagram A s χ s y A B B(s,1) PA t π C(j,1) C = A s η j χ j y A B t C C(j,1) PA \begin{matrix} & A \\ {}^s \swarrow & \overset{\chi^s}{\Leftarrow} & \searrow^{y_A} \\ B & \overset{B(s,1)}{\rightarrow} & P A \\ {}_t \searrow & \Downarrow^{\pi} & \nearrow_{C(j,1)} \\ & C \\ \end{matrix} \qquad = \qquad \begin{matrix} & & A \\ {}^s \swarrow & \overset{\eta}{\Leftarrow} & \downarrow^{j} & \overset{\chi^j}{\Leftarrow} & \searrow^{y_A} \\ B & \underset{t}{\rightarrow} & C & \underset{C(j,1)}{\rightarrow} & P A \\ \end{matrix} From Axiom 2 and the pasting lemma, we get that the LHS is an absolute left lifting (with base C(j,t)=C(j,1)tC(j,t) = C(j,1)t) if and only if η\eta is an absolute left lifting. This is exactly the realisation in our setting of the familiar correspondence between universal elements of C(j,t)C(j,t) and universal arrows from jj to tt. In particular, if π\pi is an isomorphism, then this condition holds, and we have a universal element of C(j,t)C(j,t). Note that we can understand π\pi being an isomorphism as meaning that C(j,t)C(j,t) is representable.

Now, recall the representability condition of ordinary category theory that says that a functor f:BSetf \colon B \longrightarrow \text{Set} is representable if and only if it has a universal element. We have seen that one half of this implication follows from the first two axioms of a Yoneda structure. We could take the other half as an axiom, being a basic result of category theory that we wish to incorporate into our setting.

Axiom 3*. If a 2-cell σ:A(a,1)f:APX\sigma \colon A(a,1) \Rightarrow f \colon A \longrightarrow P X yields an absolute left lifting diagram when pasted onto χ a\chi^a, then σ\sigma is an isomorphism.

Understood in our interpretation, this is indeed the other implication of the representability condition.

However this axiom is too strong to capture all the examples we want! While it does hold in ordinary category theory and in 2-categories of internal categories and variable (=indexed=parametrized) categories, it is does not hold in general in enriched category theory. For in enriched category theory, representability of a functor is expressed as an isomorphism in the base of enrichment, and the bijection of sets of the universal element condition is not enough to capture this.

Hence we do not take Axiom 3* as an axiom for Yoneda structures. Instead we take a couple of special cases. By Axiom 1 for y Ay_A and gfgf in place of aa, we have the bijections of 2-cells P1 A1 PAy Ay AC(gf,1)Pf.C(g,1)y APf.C(g,1).gf. \frac{P1_A \Rightarrow 1_{P A}}{y_A \Rightarrow y_A} \qquad \frac{C(g f,1) \Rightarrow P f.C(g,1)}{y_A \Rightarrow P f.C(g,1).g f}. Let ι A\iota_A correspond to 1 y A1_{y_A} in the first bijection, and θ f,g\theta_{f,g} correspond to the 2-cell A f B g C y A χ B(1,f) y B χ g C(g,1) PA Pf PB \begin{matrix} A & \overset{f}{\rightarrow} & B & \overset{g}{\rightarrow} & C \\ {}^{y_A} \downarrow & \overset{\chi^{B(1,f)}}{\Rightarrow} & {}^{y_B} \downarrow & \overset{\chi^g}{\Rightarrow} & \swarrow_{C(g,1)} \\ P A & \underset{P f}{\leftarrow} & P B \\ \end{matrix} in the second. These corresponding 2-cells are an identity and a pasting of two absolute left liftings respectively. Hence (by the pasting lemma in the second case) they are both absolute left liftings, and Axiom 3* would imply that they are isomorphisms. We take these special cases as our third and final axiom.

Axiom 3. The 2-cells ι A\iota_A and θ f,g\theta_{f,g} are isomorphisms.

In summary, a Yoneda structure on a 2-category consists of admissible arrows and Yoneda arrows satisfying Axioms 1, 2, and 3.

Formal category theory

Now that we have established the notion of a Yoneda structure on a 2-category, we can begin to develop category theory within such a 2-category. What is remarkable is that everything follows formally and naturally from the Yoneda structure and from the elementary two-dimesional algebra of pasting, liftings, extensions and adjunctions. In their paper, Street and Walters develop results concerning adjunctions, weighted colimits, EM objects and Kleisli objects for monads, and they introduce the notion of totality. I will now briefly survey some of these results; I refer the reader to the paper for details.

They show that the 2-categorical definition of an adjunction fgf\dashv g in terms of its unit and counit is equivalent to the condition B(f,1)A(1,g)B(f,1) \cong A(1,g), which expresses the classical hom-set definition of an adjunction. This is an isomorphism of modules, and indeed, one benefit of a Yoneda structure is that it equips our 2-category with a notion of modules (a.k.a. profunctors): modules from AA to BB are arrows APBA \longrightarrow P B.

It is well known that a satisfactory theory of colimits in enriched category theory requires the consideration of colimits weighted by presheaves. However the natural generality of the theory calls for weighting by modules (presheaves on AA are modules from II to AA). Given a module j:MPAj\colon M \longrightarrow P A and an arrow s:ACs\colon A \longrightarrow C (suitably admissible), we define the colimit colim(j,s):MC\text{colim}(j,s) \colon M \longrightarrow C of ss weighted by jj by the formula C(colim(j,s),1)(PA)(j,C(s,1))C(\text{colim}(j,s),1) \cong (P A)(j,C(s,1)). From this formula we can derive many familiar results such as the fact that left adjoints preserve colimits, associativity of colimits in the weight, and the Yoneda isomorphisms. Also we can introduce the notion of pointwise extension in terms of colimits and show that they are indeed extensions. Thus all the results of the calculus of colimits, which in enriched category theory depend on a “complicated machinery” involving enriched extranatural transformations, coends, enriched functor categories etc., in fact follow easily and formally in the context of the elementary notion of a 2-category with a Yoneda structure.

Street and Walters show that a certain result from Street’s earlier paper, the characterisation of the Eilenberg-Moore category of a monad in CAT as certain sheaves on the Kleisli category of the monad, holds for any Yoneda structure. They also show that in the presence of more properties of the 2-category (particularly a bo-ff factorisation of its arrows), we can make sense of the idea of the Kleisli object as a “full subcategory” of the Eilenberg-Moore object.

This paper introduced the notion of totality. An arrow s:ACs \colon A \longrightarrow C is said to be total when AA and ss are admissible and the module C(s,1):CPAC(s,1) \colon C \longrightarrow P A has an admissible left adjoint. It follows immediately from the developed theory that this left adjoint zz must be given by zj=colim(j,s)z j = \text{colim}(j,s) and be the pointwise left extension of ss along the Yoneda arrow y Ay_A (note that these assertions require all the admissibility assumptions in the definition). Such adjunctions are absolutely fundamental to the use of category theory; they sometimes go by the name of nerve and realization.

An object AA is said to be total when the identity arrow 1 A1_A is total; this is the same as saying the Yoneda arrow y A:APAy_A \colon A \longrightarrow P A has a left adjoint. This can be thought of as a sort of strong cocompleteness property. In particular we get a “very satisfactory” adjoint functor theorem: if AA is total, an arrow f:ABf \colon A \longrightarrow B has a right adjoint if and only if B(f,1)B(f,1) is admissible and ff preserves the colimit colim(B(f,1),1)\text{colim}(B(f,1),1).


Important motivating examples of 2-categories with Yoneda structures are 2-categories of internal categories (see the papers of Street and Weber) and enriched categories, in particular CAT is the 2-category of categories internal to SET. Given categories \mathcal{E} and 𝒱\mathcal{V} suitably nice and approriate for internalisation and enrichment respectively, we get Yoneda structures on the 2-categories Cat((\mathcal{E}) and 𝒱\mathcal{V}-Cat roughly as follows. For such a category 𝒞\mathcal{C}, the size structure arises from an object SS of 𝒞\mathcal{C} which is in some way understood as a “full subcategory” of 𝒞\mathcal{C}; for internal categories, this can be characterised as an internal full subcategory, or equivalently, a classifying discrete opfibration. Then the objects PAP A arise as exponentials S A opS^{A^{op}}, and the Yoneda arrows are exponential adjoints of hom-functors.

The 2-category LEX of finitely complete categories and finitely continuous functors inherits a Yoneda structure from CAT. This example is of interest because the total objects are (almost) Grothendieck toposes.

Another example, not treated in this paper but in this later paper of Street, are 2-categories of variable categories. These are of the form 𝒦\mathcal{K} = Hom( op\mathcal{E}^{op},CAT), consisting of pseudofunctors, pseudonatural transformations, and modifications. I mention this example here for the interesting way in which the size structure arises. An arrow f:ABf \colon A \longrightarrow B in 𝒦\mathcal{K} is admissible if for every a:UAa\colon U \longrightarrow A and b:VBb \colon V \longrightarrow B where UU and VV are representables on objects of \mathcal{E}, the comma object fa/bf a/b is representable.

Posted at March 24, 2014 5:31 AM UTC

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Re: An Exegesis of Yoneda Structures

This is awesome, Alex. Thanks a lot!

Posted by: Alex Corner on March 24, 2014 11:40 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I found it interesting to learn that the basic cells χ f\chi^f of the theory have two universal properties: an extension property from Axiom 1 and a lifting property from Axiom 2. In 1-category theory, objects with two universal properties are very special – absolute limits / colimits and finite direct sums come to mind. Are objects with two universal properties more common in 2-category theory because of the dimensional jump? In general, do nice nn-categories tend to have objects with nn universal properties (or something like that)?

Posted by: Tim Campion on March 24, 2014 4:51 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Are objects with two universal properties more common in 2-category theory because of the dimensional jump?

I wondered something similar myself. The way I explained it to myself is that the need for two universal properties here to pin down χ a\chi^a was because of the two separate notions of composition for 22-cells. Thus Axiom 1 expresses the fact that χ a\chi^a is universal with respect to vertical composition. On the other hand, Axiom 2 expresses the fact that χ a\chi^a is universal with respect to horizontal composition (or, perhaps better, with respect to whiskering). Of course by the interchange law, we can view both Axioms as diagrams involving only vertical composition. The difference is then that for Axiom 2, the diagram will involve a “whiskering variable” x:YXx \colon Y \rightarrow X, namely we can express Axiom 2 by saying that for any 22-cell X(1,x)A(a,b)X(1,x) \Rightarrow A(a,b) there is a unique 22-cell ϕ\phi such that (A(a,1)ϕ)(χ ax)(A(a,1) \circ \phi) \cdot (\chi^a \circ x) (I wish I could draw the diagrams here but I haven’t sufficiently familiarized myself with the typesetting here yet.)

But after Alexander’s comments on the right notion of universaIity for arrows in a 22-category, I’m not sure if this is a good/sensible way to look at it. But if it is, then with respect to your question

[D]o nice nn-categories tend to have objects with nn universal properties (or something like that)?

perhaps the answer is no, because even with higher nn we don’t really get extra notions of composition. For instance, my understanding is that in a Gray category we again only define vertical composition for 33-cells in the usual way and then we again have whiskerings, but this time with respect both to 11-cells and 22-cells. But perhaps these should be considered as three separate notions of composition - it’s probably very basic but I don’t know.

My thought, which I think might be related to yours, was to ask whether when we get these double universal properties then we should suspect that what we are trying to express is in fact more naturally a 33-categorical fact. The same way that, say, to say that an object in a 11-category is an absolute limit is really to try and express a fact about the 22-category CatCat. I don’t know how to precisify ”fact” here, but maybe someone can.

Posted by: Dimitris on March 24, 2014 10:36 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

That’s a very interesting observation. I can’t think of any evidence that objects with two universal properties are more common in higher dimensions, but it might be interesting to note that there is another similarity between the universal properties of representable profunctors and those in 1-category theory that you mention.

Absolute co/limits (which include finite direct sums in additive categories) are absolute (i.e. preserved by all functors) “because” they are “diagrammatically characterized”. E.g. a binary direct sum ABA\oplus B is uniquely characterized by injections i:AABi:A\to A\oplus B and j:BABj:B\to A\oplus B and projections p:ABAp:A\oplus B \to A and q:ABBq:A\oplus B \to B such that pi=1 Ap i = 1_A and qj=1 Bq j = 1_B and qi=0q i = 0 and pj=0p j = 0 and ip+jq=1 ABi p + j q = 1_{A\oplus B}. These are all equations that are preserved by any additive functor, and they imply (and, in an additive category, are implied by) the universal properties of ABA\oplus B as a binary product and a binary coproduct.

It turns out that representable profunctors are also “diagrammatically characterized” once you find the right context in which to draw the diagrams. The right context is a double category whose vertical and horizontal arrows are functors and profunctors, respectively, and the diagrams in question exhibit (co)representable profunctors as companions and conjoints of functors. These diagrams are preserved by any double functor, and they imply and are implied by their two universal properties.

Posted by: Mike Shulman on March 24, 2014 10:55 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Does this perspective help illuminate why axiom 3 *3^\ast fails to hold in certain examples?

Posted by: Emily Riehl on March 25, 2014 8:49 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Hmm, I’m not sure. There is a property of (virtual) equipments that sort of corresponds to axiom 3 *3^{\ast}, which one might call being “strongly span-generated”: mapping out of identity proarrows detects equality and invertibility of all 2-cells in the double category. I’m not sure whether this helps illuminate it any more than looking at particular examples does, though.

Posted by: Mike Shulman on March 27, 2014 6:51 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Okay, this double-categorical view is interesting! Along the lines of what Dimitris said, a companion / conjoint is determined by two equations: one horizontal, and one vertical. I wouldn’t have dared ask for a way to see these universal properties as absolute because it seems like such an open-ended question!

On the subject of double categories and equipments: is there a treatment of size issues in this context which would be comparable to the admissible arrows of a Yoneda structure? It seems to me that in anything I’ve heard about equipments, people tend to restrict to small objects so that their profunctors can compose. But if we allow virtual equipments, maybe we can also allow large objects? Actually – (in the case of Cat\mathbf{Cat}) if all modules are valued in small sets, then a category CC is locally small if and only if it has an identity module C(1,1)C(1,1), and a functor is admissible in Street & Walters’ sense iff it has a conjoint C(f,1)C(f,1). So if we want non-admissible functors to exist, then we need to allow arrows without conjoints… Is this written down somewhere?

Posted by: Tim Campion on March 25, 2014 4:41 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

You’re absolutely right that by “going virtual” we can allow large objects in an equipment. My personal inclination is to restrict the objects of such a virtual equipment to all be locally small, so that every object has an identity module (Geoff and I actually included that in the definition of “virtual equipment”) and every arrow is “admissible” and has both a companion and a conjoint. It seems to me that the main purpose of including non-locally-small objects in a Yoneda structure is so that non-small objects will still have presheaf objects, and the main reason for wanting that is so that we can define modules between them. But in an equipment, we’re already including the modules as basic data, so we don’t need every object to have a presheaf object (or, at least, not a presheaf object that classifies all modules). Since you have to go through some contortions in the enriched case to even make sense of what you mean by a “non-locally-small” VV-category, I prefer to just omit them.

Another consequence of having the modules as basic data in a virtual equipment is that you can then characterize the presheaf objects by a universal property, and show that (insofar as they exist) they satisfy the axioms of a Yoneda structure. (Conversely, the locally small objects of a Yoneda structure form a virtual equipment.) I worked all this out some time ago, but never got around to writing it down; but I snuck some of it into enriched indexed categories, which the seminar is going to be reading later on.

Posted by: Mike Shulman on March 25, 2014 4:40 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Thanks for the post! This is one of the many papers on the list that I’ve occasionally looked at but not seriously read. It’s great to have an exposition like this available.

I also increased my vocabulary just by reading the title. An exegesis is an “exposition esp. of Scripture”, says my dictionary. That’s got to be a joke (right…?), but I still find myself squirming slightly, so appalling is the idea of treating any mathematical text like Holy Scripture. Somehow it’s even worse in category theory, where a large part of the struggle is to keep everything conceptually clean and well-organized. Revisionism is an important part of that process. I’d be depressed if I thought the Kan participants were reading these papers with anything approaching reverence.

Posted by: Tom Leinster on March 24, 2014 6:59 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I consulted an expert who tells me:

  • while exegesis started as a descriptor of Biblical exposition, it now broadly refers to textual hermeneutics or literary analysis

  • even in the context of Biblical analysis, its use isn’t normative, and in particular doesn’t necessarily imply reverence

  • and as an aside, there is a long history of mathematicians incorporating “numinous thinking into their enterprise”

For the last point she mentions Euler; sadly my math history isn’t robust enough to say any more.

What I find most impressive about this post is how persuasively Alex motivates the Yoneda structure axioms. None of this - the classical Yoneda lemma as a left extension, universal arrows as left lifting, the generalized elements perspective, the motivation for axiom 3 *3^\ast and the reason for its failure in the enriched context - is in the text by the way.

I’d go so far as to recommend that henceforth everyone should read this blog post before attempting to the read the paper.

Posted by: Emily Riehl on March 24, 2014 8:14 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I’d go so far as to recommend that henceforth everyone should read this blog post before attempting to the read the paper.

Gladly! It’s always a treat to have something short and well-written available as a primer (or possibly substitute) for the paper itself, regardless of how good the latter might be.

Posted by: Tom Leinster on March 24, 2014 8:32 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Emily’s information is correct. I myself use the word “exegesis” to refer to interpretive work one undertakes to clarify difficult or abstruse texts. For example: “While the categorical community holds Lawvere’s work in the highest esteem, some of it might benefit from further exegesis.”

I think that sense of the word is quite à propos here.

Posted by: Todd Trimble on March 24, 2014 11:02 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Thanks, both. Given your usage of the word, Todd, and its history as described by Emily’s expert, I wonder whether the relevant property of scripture here is not that it’s holy but that it’s hard to understand. Anyway, must dash — off to the pub now to do some numinous hermeneutics.

Posted by: Tom Leinster on March 24, 2014 11:46 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I take the word εξηγησις\varepsilon \xi \eta \gamma \eta \sigma \iota \varsigma to mean ‘interpretation’ (literally ‘leading/guiding out’). We should not judge words just on the company they keep. And I agree with your comments on revision. I’d hate to be thought of as anything less than iconoclastic (another word with once dominantly religious meaning). And I always joke.

Posted by: Alexander Campbell on March 25, 2014 1:00 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I wonder if there might be some further interesting examples of Yoneda structures out there. One possible example on Cat\mathbf{Cat} would be to take 𝒫C\mathcal{P}C to be the free cocompletion of CC, i.e. the category of all presheaves which are small colimits of representables. I think it should satisfy Axioms 1-3 for the same reasons the usual Yoneda structure on Cat\mathbf{Cat} does. However, the notion of admissibility will have to be modified. For a functor f:ABf: A \to B to be admissible, we have to be able to construct B(f,1):B𝒫AB(f,1): B \to \mathcal{P}A, which means that for each bBb \in B we have to require the presheaf B(f,b)B(f-,b) to be a small colimit of representables. But then it’s not clear that admissible arrows form a right ideal. On the other hand, I don’t recall Street and Walters actually making use of the right ideal condition, so maybe this is not a big deal. The total categories in this case would just be the cocomplete categories.

If this example works, then probably other cocompletion 2-monads work – for instance 𝒫C\mathcal{P}C could be the cocompletion of CC under filtered or sifted colimits. In the latter case, for a small category CC with finite coproducts, 𝒫C\mathcal{P}C would be the category of models of the Lawvere theory C opC^{\mathrm{op}}, and similarly the former case is connected to locally presentable categories.

Posted by: Tim Campion on March 24, 2014 9:51 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Street and Walters don’t give any examples to show the failure of Axiom 3* for enriched categories. One place to look for an example is in the difference between colimits and weak colimits. What’s a good example of a weak colimit which is not a colimit? Here are the relevant definitions:

M colim(j,s) χ colim(j,s) y M C C(colim(j,s),1) PM C(s,1) π C(j,1) PA = M colim(j,s) η j χ j y M C C(s,1) PA C(j,1) PM \begin{matrix} & M \\ {}^{\mathrm{colim}(j,s)} \swarrow & \overset{\chi^{\mathrm{colim}(j,s)}}{\Leftarrow} & \searrow^{y_M} \\ C & \overset{C(\mathrm{colim}(j,s),1)}{\rightarrow} & P M \\ {}_{C(s,1)} \searrow & \Downarrow^{\pi} & \nearrow_{C(j,1)} \\ & PA \\ \end{matrix} \qquad = \qquad \begin{matrix} & & M \\ {}^{\mathrm{colim}(j,s)} \swarrow & \overset{\eta}{\Leftarrow} & \downarrow^{j} & \overset{\chi^j}{\Leftarrow} & \searrow^{y_M} \\ C & \underset{C(s,1)}{\rightarrow} & PA & \underset{C(j,1)}{\rightarrow} & P M \\ \end{matrix}

The above diagrams are particular instances of the bijection πη\pi \leftrightarrow \eta that Alex describes above, between the statements of Axiom 2 and Axiom 3*. The definition of a colimit colim(j,s)\mathrm{colim}(j,s) says that there should be an isomorphism π\pi as in the left diagram. This implies that η\eta exhibits colim(j,s)\mathrm{colim}(j,s) as a left lifting of jj along C(s,1)C(s,1). The latter condition is what defines a weak colimit of ss weighted by jj.

In Chapter 3.1 of Kelly’s book on enriched categories, one finds what appears to be the same definitions, and the statement that a weak colimit is a colimit when 𝒱(I,):𝒱Set\mathcal{V}(I,-): \mathcal{V} \to \mathbf{Set} is conservative (where 𝒱\mathcal{V} is the enriching category and II is the unit). But as far as I can see he doesn’t provide an example of a weak colimit which is not a colimit. To get around the conservativity requirement, the best bet would be to look for an example enriched in some category of spaces, or perhaps something like chain complexes.

Probably the conservativity condition also suffices to prove Axiom 3*, as Alex and Emily suggested in the seminar.

Posted by: Tim Campion on March 25, 2014 4:10 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Using the term “weak colimit” for this is a bit unfortunate, since often that term has a different meaning. Are you getting that from Kelly’s “weak Kan extension”?

Another place to look for examples would be CatCat. I thought I remembered Kelly having a counterexample of this sort, but I could be mixing it up with something else.

Posted by: Mike Shulman on March 25, 2014 4:31 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I agree the “weak colimit” terminology is unfortunate. It comes from the Yoneda Structures paper.

When you say matbfCat\matbf{Cat} might provide examples, I assume you mean that Cat\mathbf{Cat}-enriched categories might provide examples, since Axiom 3* holds for Set\mathbf{Set}-enriched categories so weak colimits are colimits in that case.

You’re right about the example: Kelly gives the following one in Chapter 3.7. In the 2-category generated by a 1-cell f:01f: 0\to 1 with an endomorphism 2-cell α\alpha

f 0 α 1 f \begin{matrix} & \overset{f}{\to} & \\ 0 & \Downarrow^\alpha & 1 \\ & \underset{f}{\to} & \\ \end{matrix}

the span 1111 \leftarrow 1 \to 1 exhibits 1=1×11 = 1 \times 1 in the weak sense but not the strong sense: it does the right thing to 1-cells but not to 2-cells. I’m not sure whether it’s more generally true for enriched categories that a weak (co)limit is the same thing as a (co)limit in the underlying category.

Posted by: Tim Campion on March 25, 2014 10:31 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

It would be nice to have an example of where this comes up in practice.

Posted by: Tim Campion on March 25, 2014 10:33 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Invaluable, thanks!

Posted by: Tom Hirschowitz on March 26, 2014 4:43 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Excellent post, Alexander!

I. Probably to spell out what you said in your ‘examples’ section… One could take CATCAT and (notwithstanding size issues) take PC:=Fib(C)Fun(C op,Cat)P C := Fib(C) \cong Fun(C^{op},Cat), Grothendieck fibrations over CC. For the PP on morphisms, we do it by taking the inverse image f *:Fib(D)Fib(C)f^*: Fib(D) \to Fib(C) (indeed, the same thing will appear once one proves the Axiom 1 for FibFib). One can then observe that the Axioms 1-3 (but not the stronger version of 3!) are satisfied; one might aid oneself with Angelo Vistoli’s very readable notes (there, note the fibrational version of the ‘universal element’ statement).

I also think that simplicial presheaves would also fit as an example of a Yoneda structure, however that one would not come from what Alexander described in his last section (SSetSSet is not a subcategory of CATCAT).

II. In effect, I feel a bit uneasy about the fact that a Yoneda Structure is a structure, not a property, something really internal to a category. Certainly some Yoneda structures are more preferred than the others, and have some canonicity to them, like the SetSet-presheaves in CATCAT arise as free cocompletions of (admissible) objects. As I said above, there are other examples on CATCAT which do not satisfy the special version of Axiom 3. It makes me in turn wonder the following: how unique are Axiom 3-stronger Yoneda structures, and what their existence in a 2-category KK implies about KK? Like, if KK is VCatV-Cat for some base of enrichment VV, would that imply that VV is Cartesian?

III. As a related point; when we have our usual Yoneda structure on CatCat (or VCatV-Cat), it usually comes equipped with not just a pullback f *:PBPAf^*: P B \to P A for f:ABf: A \to B, but also with both adjoints to it; I may have understood the paper really badly, but I have not seen anything lengthy said about the adjoints. Can we see them in the generality of Yoneda structure formalism, or they might not even exist (I see that the admissibility condition can restrict the ‘size’ of objects, thus harming (co)limits potentially)?

Posted by: Eduard Balzin on March 26, 2014 10:14 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I. Awesome examples!

II. Mike Shulman points out above that the Yoneda objects 𝒫C\mathcal{P}C can be defined by a universal property if you take “bimodules/profunctors” to be part of the structure. This means working with a proarrow equipment, which is more than just a 2-category. A proarrow equipment can be defined as a 2-fully faithful inclusion from a 2-category of “arrows” to a 2-category of “proarrows” such that each arrow has a right adjoint when considered as a proarrow. (This is the adjunction B(1,f)B(f,1)B(1,f) \dashv B(f,1) that Lawvere used to motivate his discussion of Cauchy completeness.) An alternative definition is as a double category, where the arrows are vertical and the proarrows are horizontal, and there are appropriate 2-cells that allow one to turn and arrow ff into proarrows B(1,f)B(1,f), B(f,1)B(f,1). In a proarrow equipment, 𝒫C\mathcal{P}C is defined by a universal property equating proarrows ACA \to C with arrows A𝒫CA \to \mathcal{P}C.

From one perspective, this method requires us to impose even more a priori structure than a Yoneda structure does. The advantage is, perhaps, that the structure imposd is more “systematic” in some sense. Anyway, it doesn’t seem to be any problem to treat any of our examples from this perspective. For example, to get 𝒫C=Fib(C)\mathcal{P}C = \mathrm{Fib}(C), we can define a proarrow ABA \to B to be a pseudofunctor B op×ACatB^{\mathrm{op}} \times A \to \mathbf{Cat}; to get simplicial presheaves it can be a functor B op×AsSetB^{\mathrm{op}} \times A \to \mathbf{sSet}.

I’d also be interested to learn more about Axiom 3 versus Axiom 3+ (I’m having trouble writing the “star” here). Alex and Emily suggested that 𝒱Cat\mathcal{V}-\mathbf{Cat} satisfies Axiom 3+ if the functor 𝒱(I,):𝒱Set\mathcal{V}(I, -): \mathcal{V} \to \mathbf{Set} reflects isomorphisms. At least this condition implies that the “weak colimits” of the paper coincide with colimits, which is an implication of Axiom 3+. So it doesn’t seem to be connected with cartesianness – both Ab\mathbf{Ab}-enrichment and \mathbb{R}-enrichment should satisfy 3+, but Cat\mathbf{Cat}-enrichment and sSet\mathbf{sSet}-enrichment don’t.

III. The adjoints to 𝒫f\mathcal{P}f are lurking in the paper – the right adjoint is referred to as f\forall f and the left adjoint as f\exists f, as in the Lawvere paper (personally, I find this easier to keep track of than a profusion of stars and shrieks and sharps and flats, but I guess it’s less standard). The right adjoint f\forall f is constructed (in Prop. 13, p. 362) as

f=(𝒫A)(B(f,1),1)\forall f = (\mathcal{P}A)(B(f,1),1)

which is analogous to the definition

𝒫f=(𝒫A)(B(1,f),1)\mathcal{P}f = (\mathcal{P}A)(B(1,f),1)

But the left adjoint f\exists f apparently can’t be constructed. Instead (on p. 372, just before Prop. 25) they call an arrow ff “kan” if 𝒫f\mathcal{P}f has a left adjoint. I think the inability to construct f\exists f might be related to another omission: there is no proof in the paper that 𝒫C\mathcal{P}C is cocomplete! They do show that if CC is small, then 𝒫C\mathcal{P}C is total (Cor. 14, p. 363) but they don’t show that total objects are cocomplete. I’d very much like to know whether or not this holds in an arbitrary Yoneda structure. I suspect it doesn’t, since Street and Walters don’t address it.

Posted by: Tim Campion on March 27, 2014 3:15 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Tim, this is amazing, thank you so much!..

Yes, I should have observed that from my own examples, being Cartesian is not related to having a 3+3+-Yoneda structure (same issues with stars here!). I join you in the question whether total objects (with respect to a Yoneda structure) have some cocompleteness properties. Your explanation actually reminded me that Giraud, in his classical 1964 treatise called “Méthode de la descente”, discusses the right adjoint for the pullback of Grothendieck fibrations, not touching the left adjoint, which would not be expressed in any good way.

Otherwise, again, thanks for the elaborate answer.

Posted by: Eduard Balzin on March 27, 2014 4:35 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Glad to be of help, Eduard!

From what you’re saying, it sounds like setting 𝒫C=Fib(C)\mathcal{P}C = \mathrm{Fib}(C) provides a good example of a Yoneda structure where the 𝒫C\mathcal{P}C’s are not cocomplete. (From what Mike says, it sounds like 𝒱Cat\mathcal{V}-\mathbf{Cat} for a complete-but-not-cocomplete 𝒱\mathcal{V} should also work.) It would be nice to have an explicit example of a diagram in a category Fib(C)\mathrm{Fib}(C) with no colimit. This might be tricky: if we looked at strict functors CCatC \to \mathbf{Cat}, then we would have a functor category and we could compute colimits pointwise in Cat\mathbf{Cat}. So the failure to have colimits will have to hinge on a strictness issue, I guess. Maybe we’ll learn something relevant when we read Kelly’s paper on 2-limits…

Posted by: Tim Campion on March 28, 2014 3:29 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

II. Thanks Tim; I also feel that a proarrow equipment is “more systematic” than a Yoneda structure. One might also argue that equipping a 2-category with a Yoneda structure involves two separate choices: first we choose the proarrows, then we choose which proarrows to regard as “small” (in order to define the presheaf objects). Equipments let us separate these two choices.

Interestingly, there is also a “canonical” way to (try to) equip a given 2-category with proarrows, using codiscrete cofibrations.

III. One reason f !f_! (or “f\exists f”) doesn’t always exist, even in the nicest examples, is that presheaf categories generally only have small colimits, but left Kan extending along f:ABf:A\to B requires colimits of the size of AA, and not all objects of a Yoneda structure are small.

There’s also, however, an issue that the axioms of a Yoneda structure don’t really require any colimits to exist, which is related to the fact that in a virtual equipment, we may not be able to compose proarrows. In particular, I believe that we can get a Yoneda structure from VV-enriched categories as soon as VV is closed and complete (this is necessary in order to construct presheaf VV-categories), but not necessarily cocomplete. This helps explain why the right adjoint f *f_* exists in general, but not the left adjoint f !f_!: the axioms of a Yoneda structure “prefer limits to colimits”. (Virtual equipments, by contrast, have no such preference: there is a virtual equipment of VV-categories even if VV is only a multicategory, without any completeness, cocompleteness, or closedness.)

Posted by: Mike Shulman on March 27, 2014 6:48 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

The axioms of a Yoneda structure “prefer limits to colimits.”

Ironically, Street and Walters define weighted colimits but not weighted limits in a Yoneda structure. The reason is that the weight for a colimit is a presheaf, which can be represented by a map into a 𝒫C\mathcal{P}C, whereas the weight for a limit is a copresheaf, which can’t be conveniently represented in the formalism. So even though more limit-like constructions exist, we can’t talk about them systematically.

Relatedly, as Alex described in his exposition, Axiom 1 is a relative version of the Yoneda lemma for copresheaves, even though we’re axiomatizing objects 𝒫C\mathcal{P}C of presheaves. The role of the Yoneda lemma for presheaves is harder to trace in the axiomatization. I think it’s used to verify Axiom 3 in Cat\mathbf{Cat}, but indirectly. I’m not sure how to state the Yoneda lemma for presheaves in a Yoneda structure – can this be done? If so, does it hold in an arbitrary Yoneda structure?

I’m not sure what to make of this weird handedness. Should I let go of the idea that we’re “axiomatizing presheaf categories”?

For that matter, I wonder whether 𝒫C=[C,Set] op\mathcal{P}C = [C, \mathbf{Set}]^{\mathrm{op}} would define an alternate Yoneda structure on Cat\mathbf{Cat}?

Posted by: Tim Campion on March 28, 2014 3:45 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Although I agree that the “handedness” is weird (and another reason I prefer equipments), I do think it should be possible to define limits in a Yoneda structure. The more general notion of weight for either a limit or a colimit is a profunctor, and in a Yoneda structure we can say that a profunctor from AA to BB is an arrow APBA \to P B. If WW is such a profunctor, then we can consider the WW-weighted colimit of a functor BCB\to C (which, if it exists, is a functor ACA\to C), or the WW-weighted limit of a functor ACA\to C (which, if it exists, is a functor BCB\to C). I don’t have time right now to puzzle out how to define the latter in a Yoneda structure, but I think it should be possible (if for no other reason than that any Yoneda structure has an underlying virtual equipment, and I know one can define weighted limits there).

One could also consider adding copresheaf-objects (or their opposites) to a Yoneda structure; they ought to be characterizable as some kind of adjoint of PP.

Posted by: Mike Shulman on March 28, 2014 4:50 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

One could also consider adding copresheaf-objects (or their opposites) to a Yoneda structure; they ought to be characterizable as some kind of adjoint of P.

Street mentions something like this in section 5 of Elementary cosmoi, though that paper precedes this one by a few years. Loosely, a pre-cosmos is a 2-category KK with objects PAP A that ‘represent’ certain 2-sided fibrations in KK; PP is then a pseudo-functor K coopKK^{coop} \to K, and a cosmos is a pre-cosmos where PP has a left adjoint P *P^\ast. Example 3 in section 7 shows that Cat with PA=[A op,Set]P A = [A^{op}, Set] and P *A=[A,Set] opP^\ast A = [A, Set]^{op} is a cosmos.

Posted by: Finn Lawler on March 28, 2014 8:24 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Ah, yes, that must be what I was dimly remembering. An “elementary pre-cosmos” in the sense of that other paper is essentially a Yoneda structure whose “profunctors” can be identified with certain two-sided discrete fibrations, right?

Posted by: Mike Shulman on March 28, 2014 11:17 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Pretty much, I think, yes; in Elementary cosmoi the fibrations that are identified with ‘profunctors’ aren’t required to be discrete, but in Cosmoi of internal categories a few years later they are.

The preface to Elementary cosmoi mentions ‘unpublished joint work with R. Walters’, undoubtedly the Yoneda structures paper, and given how closely related the two are it should be fairly straightforward to extend the notion of Yoneda structure in the same way that cosmoses extend pre-cosmoses.

Posted by: Finn Lawler on March 29, 2014 1:53 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Coming back to this almost 2 years later, in case anyone is still watching: I think it is possible to define weighted limits in a Yoneda structure even without assuming “copresheaf objects”, but only (as far as I can see) in small objects — as opposed to weighted colimits, that can be defined in any object.

Recall that the colimit of s:ACs:A\to C weighted by j:MPAj:M\to P A is an arrow colim(j,s):MCcolim(j,s) : M \to C characterized by an isomorphism

C(colim(j,s),1)PA(j,C(s,1)), C(colim(j,s),1) \cong P A(j,C(s,1)),

which requires AA, ss, MM, and jj to be admissible (e.g. if all four objects AA, CC, MM, and PAP A are admissible — where admissibility of PAP A means that AA must be small, which is sensible).

By contrast, the limit of s:ACs:A\to C weighted by j:APMj:A \to P M should be an arrow lim(j,s):MClim(j,s) : M \to C characterized by isomorphisms natural in t:MPCt:M \to P C:

𝒦(t,C(1,lim(j,s)))𝒦(j,PC(t,C(1,s))) \mathcal{K}(t,C(1,lim(j,s))) \cong \mathcal{K}(j,P C(t,C(1,s)))

This requires CC, MM, and PCP C to be admissible, and admissibility of PCP C means that CC must be small (but interestingly, it doesn’t seem that we need AA to be small any more).

Posted by: Mike Shulman on February 26, 2016 6:48 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

One way to type the star is 3^{\ast} in math mode.

Posted by: Mike Shulman on March 27, 2014 6:52 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

This is a great piece of expository Mathematics on a difficult and abstract matter. Thank you!

I only have one remark about the bibliography: I idnt see any reference to another great expository introduction (in form of a slideshow) about Yoneda structures, written by Paul-Andre Mellies. But I must confess I didn’t opened every link in the article!

Now for the questions:

Even if it is only a small detail in such a vast amount of Mathematics, I would like to go deeper in understanding nerves and their associated realization.

Weber work about categories with arity seems to suggest that there is some sort of “general theory” about nerve and realizations (but I didn’t go through the original work; I simply read Mallies’ slides): I tried to ask the Oracle (let me call this way since we seem to be into religious puns :)) here, where I try to convey my sensation that “nerves are everywhere”.

Nerves which are “coherent” with an additional structure on a category are surprisingly ubiquitous in Algebraic Topology and higher Category Theory, and yet it seems that a “general theory” of such constructions is quite elusive.

What is the state of the art about this? Can the theory of Yoneda structures help in such a taxonomic operation? Has anybody adressed this problem before (I do think so, it’s such a natural question…)? If not, let’s do this!

Posted by: Fosco Loregian on March 27, 2014 1:00 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Dear Fosco,

I cannot reply to your question, but I must say that your link leads to the nLab ‘quasi-category’ reference (which is not bad at all, but…). The correct reference is this, I suppose.


Posted by: Eduard on March 27, 2014 5:22 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Fosco, I agree this seems like an important question. I’m not sure exactly how to precisify it, though. Which aspects of the nerve construction do we want to theorize about?

  • If we have an arrow f:ABf: A \to B in a 2-category-with-Yoneda-structure, do we always call B(f,1):B𝒫AB(f,1): B \to \mathcal{P}A a “nerve” arrow? (Conversely, does a “nerve” arrow have to be of the form B(f,1)B(f,1)? I guess I’ll assume it does.)

  • Do we require B(f,1)B(f,1) to have a left adjoint “realization” functor? This condition is what Street and Walters call “totality” of the arrow ff. They talk about some consequences of totality, but not abstract conditions guaranteeing it. The classical condition should be that AA is small and BB is cocomplete, right?

  • Do we require the realization functor to be left exact? This is touched on in the Johnstone paper we’re reading next… It might be interesting to see this translated into conditions on ff.

  • Is AA required to be some version of Δ\Delta? What aspects of classical nerve constructions might we try to recover by asking AA to be Δ\Delta-like?

  • Do we want a theory that identifies a “canonical nerve functor” given BB and some extra data? This seems to be what Weber does: if BB is given as the algebras of a suitably nice monad, he finds a canonical full, dense, subcategory ABA \to B to used for a nerve. I don’t have an understanding of what this buys us. Also, it only covers certain cases like the nerve of a category – the nerve of a topological space is built from a non-full functor ΔTop\Delta \to \mathbf{Top}.

So, as I’m seeing it, the task is to answer these questions in order to fix a definition of “arrow inducing a nerve construction in a 2-category-with-Yoneda-structure”, and then to derive theorems about this notion which reproduce what is known in various examples. Is that the sort of thing you had in mind?

Posted by: Tim Campion on March 29, 2014 12:14 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures


I do not think you should restrict to left exact realisation functors. For instance, for cubical sets, even the products are not preserved. I think Δ\Delta is pretty special in the regard of having a realisation which preserves all finite limits. Some people say that Drinfeld’s paper about the geometric realisation functor helps understand more deeply why (a shame, I have not looked into it to the moment).

As for what makes an object AA look like Δ\Delta, in the case of CatCat this has been investigated from the homotopy theory perspective by Grothendieck and then formalised by Cisinski; see test category (and in particular Cisinski’s thesis) for details. However, for a general 2-category (and for a general Yoneda structure), it is a question how to generalise these results and indeed, what to expect to be proven.

See, in the case of the PSh(A)TopPSh(A) \rightleftharpoons Top type of adjunction, we actually ask whether one can make PSh(A)PSh(A) into a higher category (via Dwyer-Kan localisation) such that it is equivalent in the (,1)(\infty,1)-sense to TopTop. What sorts of questions, generalising this one, can be asked in a general 2-category?

We can think about generalising another adjunction, SSetCatSSet \rightleftharpoons Cat, to an arbitrary context; however this is harder as having a CatCat-like object in e.g. VCatV-Cat for a non-Cartesian monoidal category VV is problematic: VCatV-Cat is usually not a VV-category. There is still a separate question about the aforementioned adjunction: if in SSetCatSSet \rightleftharpoons Cat we replace CatCat by VCatV-Cat, what should we replace SSetSSet with?

Posted by: Eduard Balzin on March 29, 2014 11:37 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Which aspects of the nerve construction do we want to theorize about?

When I posed the question I was motivated by the fact that a great number of properties of NR-pairs follow from simple abstract coend-juggling (or, which is similar but more general, from universal properties of extensions)

All you say is good questions and remarks. So let me answer sistematically:

If you juggle long enough with nerves and realizations, you’ll soon notice that B(f,1)Lan fyB(f,1)\cong \mathrm{Lan}_fy, so that you can rephrase their adjointness as the strange relation Re f=Lan yfLan fy=N f Re_f = \mathrm{Lan}_yf\dashv\mathrm{Lan}_fy = N_f Properly generalized, I guess this catches entirely the paradigm since it encodes various properties of NR-pairs at the same time. I think that a pondered question can’t forgo natural examples of such thing one encounters in “Nature”; as an example, you “often” get that the nerve functor N fN_f preserve exponentials, as a simple, purely formal consequence of something you impose on ff. Should we always ask this? I think not. On the same vein, fully faithfulness of the nerve (a fact which again follows from something you impose on ff) is a fairly useful and natural thing to ask to N fN_f. Should we always ask this? I totally agree that Street and Walters are hiding us something in the paragraph about totality; I have to think about it.

Do we require the realization functor to be left exact?

Full left exactness seems too much if you consider geometric examples, but I’m not really 110% sure. In nice situations (as the geometric ones) you get that realization commutes with products; in general I think it’s fairly unnatural.

Is AA required to be some version of Δ\Delta?

Nice point, I hope it doesn’t; a mild generalization (which I think is hidden in Joyal’s jargon) should be to take AA as a geometric shape for higher structures. I saw Dimitri Ara building a full hierarchy of “generalized nerves” in a talk he held in Paris; I found extremely interesting that there’s a recognizable paradigm guiding such constructions: in the case n=2n=2 you define a functor Δ2Cat\Delta\to2-Cat sending [n][n] to the category having objects {1,,n}\{1,\dots,n\} and morphisms iji\to j are the poset P[i,j]P[i,j] of subsets containing {i,j}\{i,j\}; then you take the associated nerve et voila; the “simplicially coherent nerve” simply goes one step further from this, taking the nerve of the poset P[i,j]P[i,j].

I guess that in general you start considering functors from Joyal’s Θ n\Theta_n categories to a suitable category of nn-categories, but at the moment I don’t know how to go further.

Posted by: Fosco Loregian on March 29, 2014 12:02 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

I admit I (still) have not completely understood Weber’s work, but there are lots of examples of dense full subcategories of Eilenberg–Moore categories. For instance, if the base is locally κ\kappa-presentable and the monad is κ\kappa-accessible, then the full subcategory of the Kleisli category spanned by the objects that are κ\kappa-presentable in the base is a dense full subcategory of the Eilenberg–Moore category.

Posted by: Zhen Lin on March 29, 2014 12:45 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Okay – so I guess it’s reasonable for the moment to take “the nerve construction” to refer to the following situation: f:ABf: A \to B is an admissible arrow with admissible domain in a 2-category-with-Yoneda-structure. Then if B(f,1):B𝒫AB(f,1): B \to \mathcal{P}A has a left adjoint z:𝒫ABz: \mathcal{P}A \to B, we say that B(f,1)B(f,1) is the nerve induced by ff and zz is the realization induced by ff.

Then Street and Walters have a few things to say about the nerve construction.

  1. By Axiom 1, the nerve functor is Lan fy A\mathrm{Lan}_f y_A.

  2. Since y Ay_A is fully faithful, we have fzy Af \cong z \circ y_A canonically. The realization of the Yoneda embedding of our test objects recovers their original instantiation via ff (this comes at the top of p. 372).

  3. If AA is small and zz is admissible, then z=Lan yfz = \mathrm{Lan}_y f and the extension is even pointwise (this also comes in the first half of p. 372).

So in particular we find that if Lan fy\mathrm{Lan}_f y has a left adjoint, it’s given by Lan yf\mathrm{Lan}_y f. I should expect that the logic can be reversed to show that if Lan yf\mathrm{Lan}_y f exists and is pointwise, then it is lefta adjoint to Lan fy\mathrm{Lan}_f y.

What other important properties can be derived from these? What important properties can’t? On the other hand, is the setting too restrictive for some examples?

One thing we might like to do is to formulate conditions on f:ABf: A \to B ensuring that the realization exists. We might need to augment the theory of Yoneda structures in order to do this convincingly…

Posted by: Tim Campion on March 29, 2014 4:45 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

(I spotted two typos in the first lines but it was too late to edit… I’m really sorry!)

Posted by: Fosco Loregian on March 27, 2014 1:03 AM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Let me throw my two cents in here. As I haven’t read the Street-Walters paper nor followed all the post please forgive me if what follows is off the track and you guys are above this.

A synthetic approach to realization functors started with the Applegate and Tierney paper in here[]. This can be seen as an instance of (co)shape theory, hence the coend juggling and Kleisli objects make their appearance (so Zhen Lin’s remark should indicate a robust class of examples). It always occured to me that the cosmological environment for Morita theory has significant overlap with the requirements in order to make the Cordier-Porter shape theory work. Now shape theory has generalizations to 2-cats (Luciano Strammaccia; Betti [] 1984 shows how 2-shape invariance for rings is tied to Cauchy convergence of a profunctor) and simplicial enrichment (Cordier-Porter-Batanin), with the latter being very much one of the developments that led to n-cats, so that this should be ‘aufgehoben’ in the Joyal work on quasicats (with which I am not really familiar!) and is probably no news for someone familiar with the recent literature.

On the risk of sounding like Cato, the elder. It seems to me that it is somewhat underappreciated that Lawvere has published a 7 page appreviated version of ‘Pursuing Stacks’ in TAC 2007: cohesion is actually very much about synthetic homotopy theory! (Well, probably I shouldn’t shoot my mouth off on PS, because I never came farther than the ‘letter to Quillen’ so I can only guess what is in there.) Lawvere has thought intensively since at least the mid 1980s on these matters in constant dialogue with Grothendieck, and it is worthwhile to take into consideration his ideas on this which are somewhat spread all over his writings e.g. in a catlist remark on Emily’s work here([] sorry from here the page doesn’t seem to work- it was the comment on 2011/10/26. If I recall correctly there should also be some more explicit remarks on the exactness of realisation&how old work of Joyal shows it is not necessary- in his contributions to the list!).

Posted by: thomas holder on March 30, 2014 1:13 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Thomas, obviously your links aren’t working. Are you checking them after you’ve hit “preview”?

With the default settings, the syntax is [link text](

Posted by: Tom Leinster on March 30, 2014 4:11 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

Thanks, good to know this for the next time! I’ve tried some bracketing in order to create the hyperlinks directly but in the end it didn’t bother me much that that some one interested in a link has to copy it into his browser by hand because like that one gets an idea what and from where one downloads.

So what I meant when I wrote that the link doesn’t work is that on my screen just the frame of the catlist appears without the Lawvere message although in the search engine this seems to be the good link to the text.

By the way, my old Latin teacher called and reminded that we Romans ‘abbreviate’, sorry for that one too!

Posted by: thomas holder on March 30, 2014 6:47 PM | Permalink | Reply to this

Re: An Exegesis of Yoneda Structures

A new question which I find intriguing. I’ve absolutely no clue about the answer!

Does the 2-category of multicategories have a Yoneda structure?

Posted by: Fosco Loregian on March 31, 2014 7:29 AM | Permalink | Reply to this

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