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July 19, 2017

What is the Comprehension Construction?

Posted by Emily Riehl

Dominic Verity and I have just posted a paper on the arXiv entitled “The comprehension construction.” This post is meant to explain what we mean by the name.

The comprehension construction is somehow analogous to both the straightening and the unstraightening constructions introduced by Lurie in his development of the theory of quasi-categories. Most people use the term \infty-categories as a rough synonym for quasi-categories, but we reserve this term for something more general: the objects in any \infty-cosmos. There is an \infty-cosmos whose objects are quasi-categories and another whose objects are complete Segal spaces. But there are also more exotic \infty-cosmoi whose objects model (,n)(\infty,n)-categories or fibered (,1)(\infty,1)-categories, and our comprehension construction applies to any of these contexts.

The input to the comprehension construction is any cocartesian fibration between \infty-categories together with a third \infty-category AA. The output is then a particular homotopy coherent diagram that we refer to as the comprehension functor. In the case A=1A=1, the comprehension functor defines a “straightening” of the cocartesian fibration. In the case where the cocartesian fibration is the universal one over the quasi-category of small \infty-categories, the comprehension functor converts a homotopy coherent diagram of shape AA into its “unstraightening,” a cocartesian fibration over AA.

The fact that the comprehension construction can be applied in any \infty-cosmos has an immediate benefit. The codomain projection functor associated to an \infty-category AA defines a cocartesian fibration in the slice \infty-cosmos over AA, in which case the comprehension functor specializes to define the Yoneda embedding.

Classical comprehension

The comprehension scheme in ZF set theory asserts that for any proposition ϕ\phi involving a variable xx whose values range over some set AA there exists a subset

{xAϕ(x)}\{ x \in A \mid \phi(x)\}

comprised of those elements for which the formula is satisfied. If the proposition ϕ\phi is represented by its characteristic function χ ϕ:A2\chi_\phi \colon A \to 2, then this subset is defined by the following pullback

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of the canonical monomorphism :12\top \colon 1 \to 2. For that reason, 22 is often called the subobject classifier of the category Set\text{Set} and the morphism :12\top\colon 1 \to 2 is regarded as being its generic subobject. On abstracting this point of view, we obtain the theory of elementary toposes.

The Grothendieck construction as comprehension

What happens to the comprehension scheme when we pass from the 1-categorical context just discussed to the world of 2-categories?

A key early observation in this regard, due to Ross Street I believe, is that we might usefully regard the Grothendieck construction as an instance of a generalised form of comprehension for the category of categories. This analogy becomes clear when we observe that the category of elements of a functor F:𝒞SetF \colon \mathcal{C} \to \text{Set} may be formed by taking the pullback:

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Here the projection functor on the right, from the slice */Set{}^{\ast/}\text{Set} of the category of sets under the one point set, is a discrete cocartesian fibration. It follows, therefore, that this pullback is also a 2-pullback and that its left-hand vertical is a discrete cocartesian fibration.

Street’s point of view is (roughly) that in a 2-category 𝒦\mathcal{K} it is the (suitably defined) discrete cocartesian fibrations that play the role that the sub-objects inhabit in topos theory. Then the generic sub-object :1Ω\top\colon 1\to \Omega becomes a discrete cocartesian fibration :S *S\top\colon S_\ast\to S in 𝒦\mathcal{K} with the property that pullback of \top along 1-cells a:ASa\colon A\to S provides us with equivalences between each hom-category Fun 𝒦(A,S)\text{Fun}_{\mathcal{K}}(A,S) and the category dCoCart(𝒦) /A\text{dCoCart}(\mathcal{K})_{/A} of discrete cocartesian fibrations over AA in 𝒦\mathcal{K}.

This account, however, glosses over one important point; thus far we have only specified that each comparison functor Fun 𝒦(A,S)dCoCart(𝒦) /A\text{Fun}_{\mathcal{K}}(A,S) \to \text{dCoCart}(\mathcal{K})_{/A} should act by pulling back :S *S\top\colon S_{\ast}\to S along each 1-cell a:ASa\colon A\to S. We have said nothing about how, or weather, this action might extend in any reasonable way to 2-cells ϕ:ab\phi\colon a\Rightarrow b in Fun 𝒦(A,S)\text{Fun}_{\mathcal{K}}(A,S)!

The key observation in that regard is that for any fixed “representably defined” cocartesian fibration p:EBp\colon E\to B in a (finitely complete) 2-category 𝒦\mathcal{K}, we may extend pullback to define a pseudo-functor Fun 𝒦(A,B)𝒦/A\text{Fun}_{\mathcal{K}}(A,B)\to\mathcal{K}/A. This carries each 1-cell a:ABa\colon A\to B to the pullback p a:E aAp_a\colon E_a\to A of pp along aa and its action on a 2-cell ϕ:ab\phi\colon a\Rightarrow b is constructed in the manner depicted in the following diagram:

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Here we make use of the fact that p:EBp\colon E\to B is a cocartesian fibration in order to lift the whiskered 2-cell ϕp a\phi p_a to a cocartesian 2-cell χ\chi. Its codomain 1-cell may then be factored through E bE_b, using the pullback property of the front square, to give a 1-cell E ϕ:E aE bE_{\phi}\colon E_a\to E_b over AA as required. Standard (essential) uniqueness properties of cocartesian lifts may now be deployed to provide canonical isomorphisms E ψϕE ψE ϕE_{\psi\cdot\phi}\cong E_{\psi}\circ E_{\phi} and E id aid E aE_{\id_a}\cong\id_{E_a} and to prove that these satisfy required coherence conditions.

It is this 2-categorical comprehension construction that motivates the key construction of our paper.

Comprehension and 2-fibrations

In passing, we might quickly observe that the 2-categorical comprehension construction may be regarded as being but one aspect of the theory of 2-fibrations. Specifically the totality of all cocartesian fibrations and cartesian functors between them in 𝒦\mathcal{K} is a 2-category whose codomain projection coCart(𝒦)𝒦\text{coCart}(\mathcal{K})\to\mathcal{K} is a cartesian 2-fibration, it is indeed the archetypal such gadget. Under this interpretation, the lifting construction used to define the pseudo-functor Fun 𝒦(A,B)𝒦 /A\text{Fun}_{\mathcal{K}}(A,B) \to \mathcal{K}_{/A} is quite simply the typical cartesian 2-cell lifting property characteristic of a 2-fibration.

In an early draft of our paper, our narrative followed just this kind of route. There we showed that the totality of cocartesian fibrations in an \infty-cosmos could be assembled to give the total space of a kind of cartesian fibration of (weak) 2-complicial sets. In the end, however, we abandoned this presentation in favour of one that was more explicitly to the point for current purposes. Watch this space, however, because we are currently preparing a paper on the complicial version of this theory which will return to this point of view. For us this has become a key component of our work on foundations of complicial approach to (,)(\infty,\infty)-category theory.

An \infty-categorical comprehension construction

In an \infty-cosmos 𝒦\mathcal{K}, by which we mean a category enriched over quasi-categories that admits a specified class of isofibrations and certain simplicially enriched limits, we may again define p:EBp \colon E \twoheadrightarrow B to be a cocartesian fibration representably. That is to say, pp is a cocartesian fibration if it is an isofibration in the specified class and if Fun 𝒦(X,p):Fun 𝒦(X,E)Fun 𝒦(X,B)\text{Fun}_{\mathcal{K}}(X,p) \colon \text{Fun}_{\mathcal{K}}(X,E) \to \text{Fun}_{\mathcal{K}}(X,B) is a cocartesian fibration of quasi-categories for every \infty-category XX. Then a direct “homotopy coherent” generalisation of the 2-categorical construction discussed above demonstrates that we define an associated comprehension functor:

c p,A:Fun 𝒦(A,B)coCart(𝒦) /A.c_{p,A} \colon \mathfrak{C}\text{Fun}_{\mathcal{K}}(A,B)\to \text{coCart}(\mathcal{K})_{/A}.

The image lands in the maximal Kan complex enriched subcategory of the quasi-categorically enriched category of cocartesian fibrations and cartesian functors over AA, so the comprehension functor transposes to define a map of quasi-categories

c p,A:Fun 𝒦(A,B)(coCart(𝒦) /A)c_{p,A} \colon \text{Fun}_{\mathcal{K}}(A,B) \to \mathbb{N}(\text{coCart}(\mathcal{K})_{/A})

whose codomain is defined by applying the homotopy coherent nerve.

Straightening as comprehension

The “straightening” of a cocartesian fibration into a homotopy coherent diagram is certainly one of early highlights in Lurie’s account of quasi-category theory. Such functors are intrinsically tricky to construct, since that process embroils us in specifying an infinite hierarchy of homotopy coherent data.

We may deploy the \infty-categorical comprehension to provide a alternative approach to straightening. To that end we work in the \infty-cosmos of quasi-categories qCat\text{qCat} and let A=1A=1, and observe that the comprehension functor c p,1:BqCatc_{p,1}\colon \mathfrak{C}B \to \text{qCat} is itself the straightening of pp. Indeed, it is possible to use the constructions in our paper to extend this variant of unstraightening to give a functor of quasi-categories:

(coCart /B)Fun(B,Q)\mathbb{N}(\text{coCart}_{/B}) \to \text{Fun}(B,Q)

Here QQ is the (large) quasi-category constructed by taking the homotopy coherent nerve of (the maximal Kan complex enriched subcategory of) qCat\text{qCat}. So the objects of Fun(B,Q)\text{Fun}(B,Q) correspond bijectively to “straight” simplicial functors BqCat\mathfrak{C}B\to\qCat. We should confess, however, that we do not explicitly pursue the full construction of this straightening functor there.

Unstraightening as comprehension

In the \infty-categorical context, the Grothendieck construction is christened unstraightening by Lurie. It is inverse to the straightening construction discussed above.

We may also realise unstraightening as comprehension. To that end we follow Ross Street’s lead by taking Q *Q_{\ast} to be a quasi-category of pointed quasi-categories and apply the comprehension construction to the “forget the point” projection Q *QQ_{\ast}\to Q. The comprehension functor thus derived

c p,A:Fun(A,Q)(dCoCart /A)c_{p,A} \colon Fun(A,Q) \to \mathbb{N}\left(dCoCart_{/A}\right)

defines a quasi-categorical analogue of Lurie’s unstraightening construction. In an upcoming paper we use the quasi-categorical variant of Beck’s monadicity theorem to prove that this functor is an equivalence. We also extend this result to certain other \infty-cosmoi, such as the \infty-cosmos of (co)cartesian fibrations over a fixed quasi-category.

Constructing the Yoneda embedding

Applying the comprehension construction to the cocartesian fibration cod:A 2Acod : A^2 \to A in the slice \infty-cosmos 𝒦 /A\mathcal{K}_{/A}, we obtain a map

y:Fun 𝒦(1,A)Cart(𝒦) /Ay \colon \text{Fun}_{\mathcal{K}}(1,A) \to \mathbb{N}Cart(\mathcal{K})_{/A}

that carries an element a:1Aa \colon 1 \to A to the groupoidal cartesian fibration dom:AaAdom : A \downarrow a \to A. This provides us with a particularly explicit model of the Yoneda embedding, whose action on hom-spaces is easily computed. In particular, this allows us to easily demonstrate that the Yoneda embedding is fully-faithful and thus that every quasi-category is equivalent to the homotopy coherent nerve of some Kan complex enriched category.

Posted at July 19, 2017 2:13 AM UTC

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Re: What is the Comprehension Construction?

Most people use the term \infty-categories as a rough synonym for quasi-categories […]

I find that tragic, since quasi-categories only model a very special class of \infty-categories, namely those whose jj-morphisms are equivalences for j>1j &gt; 1. The correct rough name for something like that is (,1)(\infty,1)-category.

(You know this, of course, but there could be impressionable children in the room, and I don’t want them to grow up confused.)

[…] but we reserve this term for something more general: the objects in any \infty-cosmos.

Is there a known \infty-cosmos of actual \infty-categories, without any restrictions like “all jj-morphisms are equivalences for j>1j &gt; 1”?

But there are also more exotic \infty-cosmoi whose objects model (,n)(\infty,n)-categories […]

That’s better. For the incognoscenti, that means “\infty-categories where all jj-morphisms are equivalences for j>nj &gt; n”. So I’m essentially asking if it’s known that we can take the limit nn \to \infty and get this to still work.

Or even better: is there an \infty-cosmos of small \infty-cosmoi? That’s when I’d get really happy: when the formalism is general enough to eat itself.

Posted by: John Baez on August 1, 2017 5:17 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

We do have an \infty-cosmos whose objects are (,)(\infty,\infty)-categories, with possibly non-invertible morphisms all the way up. Dominic Verity has established a variety of model structures on the category of stratified simplicial sets whose fibrant/cofibrant objects are some variety of complicial sets (nee weak complicial sets). One of these recovers the Joyal model structure for quasi-categories, which are precisely the saturated complicial sets with every simplex above dimension 1 marked.

My favorite of Dom’s model structures is the model structure for saturated complicial sets, those complicial sets in which all equivalences are marked. This model category is enriched over the Joyal model structure by taking the (,1)(\infty,1)-categorical “cores” of the internal hom objects, and thus defines an \infty-cosmos whose objects are (,)(\infty,\infty)-categories.

Some more details can be found in lecture notes for a mini course on complicial sets I gave at the Matrix Institute in June 2016.

Posted by: Emily Riehl on August 3, 2017 11:48 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

complicial sets (nee weak complicial sets)

Is this an official proposal to change the terminology? It’s probably about time, given that the default meaning of “nn-category” these days is weak.

Posted by: Mike Shulman on August 3, 2017 5:02 PM | Permalink | Reply to this

Re: What is the Comprehension Construction?

Yes, I officially propose using complicial sets for what were originally called “weak complicial sets” and strict complicial set for what were originally called “complicial sets.”

Dom and I have also discussed using the term nn-complicial set to mean a stratified simplicial set which is

  • a (weak) complicial set
  • that is nn-trivial, meaning that every kk-simplex with k>nk \gt n is marked,
  • and also saturated, meaning that all equivalences are marked.

Then a 0-complicial set is just a Kan complex (with the maximal marking), a 1-complicial set is just a quasi-category (with its natural marking), and the nn-complicial sets are conjecturally a model for (,n)(\infty,n)-categories satisfying the axioms of Barwick and Schommer-Pries.

Posted by: Emily Riehl on August 4, 2017 8:35 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

By the way, will those lecture notes be in the Matrix annual book that is—apparently—being published next month? (My only source for the timing is the Gazette of the AustMS, nothing from Springer who is actually doing the publishing.)

Posted by: David Roberts on August 4, 2017 8:01 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

Yes I believe so. They are part of the proceedings for the first ever Matrix workshop, organized by Marcy Robertson and Phil Hackney. Ben Ward, who served as the editor for the proceedings volume for our workshop, would know.

Posted by: Emily Riehl on August 4, 2017 8:37 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

Cool, thanks. I was dealing with him too, I’ve emailed to see if I have the latest information.

Posted by: David Roberts on August 6, 2017 11:37 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

I recall sending someone - I think you, though possibly Mike - my first guest blog post written back when I was an impressionable child and being told that nn-Category Café always utilizes the proper terminology: (,1)(\infty,1)-categories instead of “\infty-categories.”

Two summers ago after a lot of thought I suggested to Dom that we re-appropriate the word “\infty-categories” to use it in the way we do in this post: to mean an object in any \infty-cosmos. It’s true that this usage of “\infty-categories” doesn’t clarify what sort of infinite-dimensional category is meant - but neither do the common ones. And it allows us to state our theorems in an elegant way that makes their essential content relatively accessible to new readers (and importantly also to google), while remaining within a fully precise (i.e., not hand wavy) axiomatic framework.

So, for example, “any equivalence between \infty-categories can be promoted to an adjoint equivalence” means that in any \infty-cosmos, any equivalence (where an equivalence is defined to be a map inducing equivalences on the function complexes) may be made into an adjoint functor (where an adjunction is defined to be an adjunction in the homotopy 2-category of the \infty-cosmos).

Sometimes while introducing this point of view, we invite our readers to pretend that we are using “\infty-categories” to mean only “well-behaved models of (,1)(\infty,1)-categories,” namely quasi-categories, complete Segal spaces, Segal categories, or saturated 1-complicial sets. For instance, we adopt this tact in the open pages of \infty-category theory from scratch. But in developing our formal theory of (,1)(\infty,1)-categories, we frequently take advantage of induced \infty-cosmos structures, e.g., on the cartesian fibrations over a fixed base \infty-category in a given \infty-cosmos. So our “\infty-categories” appearing for instance in the theorem mentioned above may be fibered as well.

Posted by: Emily Riehl on August 3, 2017 12:30 PM | Permalink | Reply to this

Re: What is the Comprehension Construction?

Appropriating the word “\infty-category” is a cute idea. I’ve been tempted to do something similar myself, appropriating the word “category” to refer to an object of a (sufficiently highly-structured) formal category theory (proarrow equipment, Yoneda structure, etc.). But your choice to do this seems to confuse many people, and I have trouble remembering myself that you’re using it in a much wider sense than Lurie’s school.

I expect the reason it’s confusing is that elsewhere in category theory, when generalizing from a particular categorical structure to a general class of them, we don’t usually keep the particular name of the objects. For instance:

  • Topoi are a generalization of SetSet, but we don’t call the objects of an arbitrary topos “sets”. (Lawvere has proposed calling the objects of an arbitrary topos “variable sets”, but this doesn’t seem to have caught on very widely.)
  • Abelian categories are a generalization of AbAb, but we don’t call the objects of an arbitrary abelian category “abelian groups”.
  • 2-categories are a generalization of CatCat, but we don’t call the objects of an arbitrary 2-category “categories”. Same for proarrow equipments, fibrational cosmoi in Street’s sense, etc.
  • (,1)(\infty,1)-topoi (in the usual sense) are a generalization of Gpd\infty Gpd, but we don’t call the objects of an arbitrary (,1)(\infty,1)-topos “\infty-groupoids”.
  • Stable (,1)(\infty,1)-categories are a generalization of SpSp, but we don’t call the objects of an arbitrary stable (,1)(\infty,1)-category “spectra”.

The only situation I can think of in which we sometimes do something like this is when working in the internal logic of such a class of categorical structures. In the internal logic of a topos, for instance, the objects really do behave like sets (they have “elements” and are “determined by their elements”), and we often call them “sets” (though “types” is also common). In fact, I often find this linguistic distinction useful to help keep track of whether I’m working in the internal language of a category or externally using its categorical structure.

Posted by: Mike Shulman on August 3, 2017 5:00 PM | Permalink | Reply to this

Re: What is the Comprehension Construction?

This is a very interesting comment:

The only situation I can think of in which we sometimes do something like this is when working in the internal logic of such a class of categorical structures. In the internal logic of a topos, for instance, the objects really do behave like sets (they have “elements” and are “determined by their elements”), and we often call them “sets” (though “types” is also common). In fact, I often find this linguistic distinction useful to help keep track of whether I’m working in the internal language of a category or externally using its categorical structure.

Thank you for sharing it.

I agree that our usage of “\infty-category” is confusing, particularly because it’s both more restrictive than the common usage (in that only the well-behaved models of (,1)(\infty,1)-categories are examples) and also more general (in that not all \infty-categories are (,1)(\infty,1)-categories).

But I haven’t been able to come up with any other strategy, aside from this cheeky terminological re-appropriation, for achieving its purpose: I want people who are searching for information about \infty-categories (however they mean this) to be able to discover without necessarily being told that \infty-cosmoi might be relevant. It seems tricky to publicize our complementary approach to the Joyal-Lurie school of (,1)(\infty,1)-category theory without using some of the same terminology.

Now if the notion of \infty-cosmoi or something like it becomes as ubiquitous as some of the categories you listed above, then I agree it would no longer be necessary to give a special name to their objects, but I’m not counting on this ever coming to pass…

Posted by: Emily Riehl on August 4, 2017 8:49 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

That’s a reasonable point.

Posted by: Mike Shulman on August 7, 2017 11:13 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

Emily wrote:

Two summers ago after a lot of thought I suggested to Dom that we re-appropriate the word “\infty-categories” to use it in the way we do in this post: to mean an object in any \infty-cosmos.

I don’t mind that as long as this includes some version of what I call \infty-categories. Apparently it does, since you say:

We do have an \infty-cosmos whose objects are (,)(\infty,\infty)-categories, with possibly non-invertible morphisms all the way up.

In general, I guess I don’t mind stretching terminology, but I do mind squashing it. When someone says they’ve proved something about \infty-categories, I’d be happy to know they proved something more general than I thought, but annoyed if they only proved it only for (,1)(\infty,1)-categories.

It may take a few decades, or even a century, for the notation to settle down in higher category theory. When the time is ripe, we should work very hard to get it right. But the time is not ripe.

Posted by: John Baez on August 6, 2017 10:52 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

It may take a few decades, or even a century, for the notation to settle down in higher category theory. When the time is ripe, we should work very hard to get it right. But the time is not ripe.

Isn’t it usually too late to change the notation and terminology after a subject has been established for decades or centuries?

Posted by: Mike Shulman on August 7, 2017 11:14 AM | Permalink | Reply to this

Re: What is the Comprehension Construction?

Isn’t it usually too late to change the notation and terminology after a subject has been established for decades or centuries?

I think if you look at old mathematics you’ll see it was written in ways that are quite hard to understand now. So, there seems to be room for quite a bit of change over time.

Anyway, go ahead and try to perfect the terminology — I shouldn’t try to stop you! Just don’t expect it’ll settle down soon.

One reason is that when dealing with higher categorical structures, mathematicians need time to get used to the idea that any concept is a point in some space of concepts, so we should have separate names for individual concepts and connected components of this space. For example, sometimes we want to use the word ‘(,1)(\infty,1)-category’ to mean either a quasicategory or a simplicially enriched category or… various other equivalent concepts. But sometimes, in a specific conversation or text, we want to use it as a shorthand for one of these specific concepts. And we’re not terribly graceful, yet, at negotiating the switch between these two different attitudes.

It’s really just distinguishing between points in some space and connected components of that space, so mathematicians should be able to handle it. But right now they tend to want to use the same name for a connected component and a specific ‘basepoint’ of that connected component — while simultaneously feeling guilty for doing this, or feeling like it’s a temporary problem and we’ll eventually choose the ‘best’ basepoint, like ‘the’ definition of nn-categories. I guess this is because most mathematicians are still more used to subjects where the concepts involved are closer to forming a discrete space, or set.

In short, the homotopification of mathematics also applies to terminology, and this will take a while to deal with.

But maybe youngsters like you are already completely used to this! So maybe it’ll just take time for the old folks to die off.

Posted by: John Baez on August 8, 2017 12:05 PM | Permalink | Reply to this

Re: What is the Comprehension Construction?

We’re probably going to call the real ones weak omega-categories, because infinity, infinity is a mouthful. Specific models will probably be called omega-quasicategories (Theta quasicategories) and probably omega-segal spaces (resp. Theta-segal spaces).

I hope by this time next year we will have some progress on a Yoneda theorem, after which it should be lots of open country. We already have a lot of fun lax machinery that doesn’t exist in the ordinary case. Will keep you abreast of it all. Comprehension here might get us just enough to get Yoneda, but it only gets us presheaves of spaces, not full cartesian fibrations (presheaves of weak omega-categories).

Posted by: Harry on August 11, 2017 4:17 PM | Permalink | Reply to this

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