### Intrinsic Naturalness

#### Posted by Urs Schreiber

Lately there have been remarkable developments in higher category theory.

What used to have a touch of alchemy to it – in its mystery, its grand hopes, its plethora of recipes tried out in hard and lonely work in long nights – is becoming chemistry: what used to be hypothesis and conjectures have become theorems; what used to be a philosopher’s stone seemingly out of reach has become a tangible jewel that you can touch, hold against the light, marvel at – and finally use to cut through the glass roof that has been impeding progress for so long: higher category theory.

Or so some think. Discussion with colleagues reveals that the perception of and feelings about what has been achieved – and is being achieved as we speak – varies. While parts of the community are storming ahead with the new technology that has become available, in other parts reservation and scepticism towards this activity is being felt.

Is this really the philosopher’s stone that the search was after?
Isn’t its shiny appearance a cheap trick achieved by taking
that old pebble called *homotopy theory*, and polishing it a bit?
Is what superficially looks impressive rather intrinsically a kludge geared
to serve a purpose and good enough to impress the mundane, but far
from being the natural god-given structure that the inner circle
of researchers in higher category theory knows – and rightly knows! –
is the true goal of the search?

This are, in impressionistic paraphrase, questions being asked behind the scenes. As I have realized in long private discussions recently. And these are good and important questions. If anything as important as a first working implementation of higher category theory is being claimed – explicitly or not – , the claim deserves to be carefully scrutinized.

But careful scrutiny requires an effort to obtain a clear picture of the situation to the same extent that it requires a critical mindset. Imagine the alchemist producing finally the philosopher’s stone – and then discarding it onto the heap of failed attempts for not recognizing it.

Recently I was standing in an alchemist friend’s laboratory and we were looking at that stone. My colleague pointed at it and exclaimed: “Look, it is not natural. This cannot be the answer.” To which I replied: “But wait, you are looking at it as we use it to cut through that glass. Pick it up instead and hold it against the sunlight, so that you see its intrinsic colors, not the reflection of the workbench.” I picked up the stone and held it against the light in three different angles, and we were bathed in its light.

My friend agreed that he hadn’t looked at the stone in this light, and that this did make a difference. He then asked me to share the view from these three angles here on the blog. Which is what I now do.

When you are standing at the workbench, it may seem as if the new
developments in higher category theory – in $(\infty,1)$-category theory
notably – are based on unnatural kludges from old toolboxes such as notably
those tools called *model categories* and *triangulated categories* .
The impression might arise that the presently active approach is suggesting to
fundamentally *base* higher category theory on model category theory!

But notice that the opposite is true.

Model categories and triangulated categories are toolboxes that have been used for a long time and of which it has always been clear that

they work – they do capture aspects of nature;

they are kludgy and unnatural and at best shadows of a deeper natural structure.

This kind of situation is the killer application for category theory: any field in math full of interesting results but with conceptually awkward foundation. Usually this is a sign that a nice and powerful category theoretic conceptualization is waiting to be uncovered, one that makes all the concepts transparent and makes all the complexity manifestly follow by a systematic process from simple natural basic structures.

And I am claiming that it is this kind of *abstract category theory success* that we
are currently witnessing. If you think that $(\infty,1)$-category theory
is done all with model categories, then you have never tried to read the literature:

after the relation to model category theory is made crystal clear in the
theorem that combinatorial model categories are precisely a generators-and-relations
presentation of locally presentable $(\infty,1)$-cartegories in *Higher Topos Theory*,
they are mostly entirely dropped in favor of intrinsic $(\infty,1)$-category theory.
There is hardly a model category model being used as a computational tool in the
further development of the theory. Much to the regret of poor souls like me!
An explicit model category model of some $(\infty,1)$-category helps me handle
these beasts. It is like choosing a basis for my vector spaces often helps me
to handle these. Even though I know that theoretically I should treat them as
abstract vector spaces, sometimes an elegant, abstract, intrinsic proof is
harder to come by then a kludgy pedestrian component-ridden proof. This
fault of a fallible man is not a fault of the theory, though.

I claim it is the other way round: researchers have used and developed model category tools for ages to achieve their goals, thereby producing impressive but – to the eyes of the abstract category theorist or toposopher – unnatural-looking theories. What the present development in higher category theory does achieve is a radical re-conceptualization of these workbench scenarios: it unravels the intrinsic nature underlying these constructions.

Here are three main examples.

**$(\infty,1)$-toposes**If we count Kenneth Brown’s 1973 work as one of the earliest publications where the central idea is put forward, there is now a history of over 35 years of published work on model category structures on simplicial presheaves. Joyal famously suggested this in his letter to Grothendieck as a possible solution to the pursuit of $\infty$-stacks. Later Jardine intensively developed the theory both in depth and in width. Toën and Vezzosi then generalized the definition to “higher categorical” sites – modeled in turn as model-category sites! The implict claim has all along been that this model structure somehow models higher Grothendieck toposes of $(\infty,1)$-sheaves/$\infty$-stacks. Somehow!

While it could be shown that this was indeed the case in lowest degree, where one could compare with the available intrinsic 2-categorical definition of stack-2-toposes, much beyond that no intrinsic higher category theory was available to check against. And wasn’t it a dubious claim? The first thing most textbooks will tell their readers about stacks is that these are pseudofunctors. But a simplicial presheaf is a

*strict*functor! How on earth can that model an $\infty$-stack? And how on earth is something as intrinsically intrinsic as topos theory to be incarnated in its higher categorical version in the usual kludgy collection of factorization systems and weak equivalences that the model category structure is based on?All this is clarified now. Now with HTT an entirely intrinsic, abstract, pure higher category theory definition of $\infty$-stack has been given: a Grothendieck $(\infty,1)$-topos of $\infty$-stacks is a reflective $\infty$-subcategory of an $(\infty,1)$-presheaf category

$Sh_{(\infty,1)}(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.$

This is as abstractly category-theoretic and topos-theoretic as it gets. This is as natural and elegant as it gets. If only we can make sense of what all these $(\infty,1)$-categorical notions mean (adjoint, limit, etc.) then this must be the right and good answer. And now we can make sense of it.

And then there is a theorem: the model category structures on simplicial presheaves that have been

*guessed*for over 35 years, effectively, to model $\infty$-stack categories do precisely this: every abstractly defined $\infty$-stack $(\infty,1)$-topos $Sh_{(\infty,1)}(C)$ has a generators-and-relation presentation by simplicial presheaves on $C$, and every such presentation presents an $\infty$-stack $(\infty,1)$-topos. So what we have now in hand is – on top of a mighty powerful computational technology that has been around all along – an understanding of the entirely natural, abstract and elegant underlying nature of the phenomenon described by this technology.In 1972 Dan Quillen – famously the inventor of the model category toolset and the advisor of Ken Brown with whom the above story began – had studied homotopy fibers of maps between nerves of categories and remarked:

Someday these ideas will undoubtedly be incorporated into a general homotopy theory for topoi.

This has now become literally true.

For an attempt at an exposition surveying these ideas see

**stable $(\infty,1)$-categories**If model categories feel bad in their component-ridden non-intrinsicness, then

*triangulated categories*are even worse. While a powerful toolset that has allowed people to venture deep into the heart of homological algebra, dissatisfaction with this toolset has been felt and expressed early on. It was an eduring source of embarrassment how ill-behaved triangulated categories were under various natural operations. Cures for this were put forward, such as notably enhanced pretriangulated dg-categories, but their definition certainly does not manifestly make the concept come closer to the natural intrinsic category theoretic definition that the pure higher category theorist would hope for.This has changed now. There is a beautiful, simple abstract concept that explains it all: that of a stable $(\infty,1)$-category.

In every $(\infty,1)$-category $C$ with a 0-object, there are two god- given operations: the formation of loop space objects and that of suspension objects

$\Omega :C \stackrel{\leftarrow}{\overset{\simeq}{\to}} C : \Sigma \,.$

These are nothing but the pullback and pushforward along itself of the inclusion of the zero object into any other object. An $(\infty,1)$-category is

*stable*if it is stable under these operations: if $\Omega$ and $\Sigma$ exhibit an auto-equivalence.That’s a one-line definition. And one whose consideration is compelling. There is no doubt that in the abstract platonic world of pure higher category theory, stable $(\infty,1)$-categories deserve to be honored for their existence.

And then there is a theorem: when you apply force and crush down a nice and elegant stable $(\infty,1)$-category to a plain old 1-category by decategorifying it against its will, what you are left with is an ill behaved totally unnatural remnant. All it remembers is a faint shadow of the simple abstract definition above: the shadow of the suspension $\infty$-functor becomes known as the shift functor, and what are known as the

*distinguished triangles*that give the notion of triangulated categories its name is nothing but a shadow of the long fibration sequences that used to exist in the stable $(\infty,1)$-category before it was forcefully decategorified.If you never liked triangulated categories, now is the time to rejoice: you will never ever have to mention them again. Use stable $(\infty,1)$-category theory instead. This is the right notion. Everything else is a kludge. (Useful as that may be for computations, sometimes.)

For more details on this see the discussion and links given at

**$(\infty,1)$-geometry**The new development in higher category theory is notably driven by the desire for a deeper understanding of generalized geometry.

Category theory and topos theory is coming back to its origins here: originally Grothendieck had recognized forcefully that category theory and topos theory provide the right language to speak about the generalized geometry that he was looking at, which happened to be that of generalized spaces modeled on formal duals to commutative rings. This single application remarkably gave rise to a huge development of abstract category theory and topos theory, which in turn provided more tools for these applications.

More recently it became very clear that an entirely higher categorical refinement of all this should not only be thinkable, but is forced upon us by natural examples: the Goerss-Hopkins-Miller theorem shows that the moduli space of derived ellptic curves seems to behave like a higher categorical version of a ringed space, whose structure sheaf consists not of rings of functions, but of $E_\infty$-rings of functions. That this moduli space

*wants*to be regarded as an object in higher categorical geometry – in derived geometry – becomes quite clear. The big challenge is to make manifest all the required abstract structure that makes this realization not only desireable, but precise and provable.This is a major motivating example behind derived geometry – higher categorical geometry; higher topos theory.

There have been early attempts to realize this. Notably Toën and Vezzosi constructed an impressive machinery for the description of derived geometry: all entirely built from the model category tool set. They generalize sites to model-sites, sheaves to objects in the model structure on simplicial presheaves on model sites and so on and so forth. An admirable undertaking. But the abstract category theorist who already feels personally hurt by the concept of model category alone, what should he or she think of this? Is such a component-ridden construction supposed to be a realization of intrinsic natural higher category theoretic geometry?

It is a shadow of it. Inspired by these developments, Lurie develops in

*Structured Spaces*the picture of a theory of general abstract higher geometry of charming natural intrinsic beauty. Don’t be fooled by the title of the series: what is being exposed here goes way, way beyond a generalization of the field of algebraic geometry. This is a text with concepts of largely the same intrinsic depth as, say, the*Sketches of an Elephant*.The simple central idea is that a concrete generalized space is an $(\infty,1)$-topos equipped with a structure sheaf $\mathcal{O}$, modeled as an $\infty$-stack-valued $\infty$-copreshaef on the $\infty$-category $\mathcal{G}$ of spaces on which the geometry is modeled

$\mathcal{O} : \mathcal{G} \to Sh_{(\infty,1)}(X)\,.$

And as shown there, by making concrete choices for that category of test spaces and then turning the crank, the abstract formalism spits out first standard algebraic geoemtry, then algebraic stack theory, then differential geometry, then derived differential geometry, derived $E_\infty$-spectral geometry and everything else.

An attempt to survey and highlight the central abstract conceptual higher categorical ideas here is at

These are three examples for the intrinsci naturalness exhibited by recent developments in higher category theory. Instead of being based on kludgy toolsets that have been around for decades, the development shows abstract category theory at its best by unifying all these toolsets into presentations for natural general abstract concepts. And then using the clarity gained this way, the theory goes well beyond what has previously been conceivable.

Or so I think. Hopefully we can discuss this here further.

## Re: Intrinsic naturalness

Urs, in your little history of “$(\infty,1)$-topos”, I can’t help but think that Dan Quillen should be mentioned somehow. In the introduction to his famous paper “Higher Algebraic K-theory I” (1972), Quillen (who was Ken Brown’s advisor) devotes a paragraph giving an apology for his proofs of Theorems A and B in that paper (these give sufficient conditions for computing the homotopy fiber of a map between nerves of categories). After discussing various techniques which might be used to prove these, he closes with:

“The present approach, based on the Dold-Thom theory of quasi-fibrations, is quite a bit shorter than the others, although it is not as clear as I would have liked, since the main points are in the references. Someday these ideas will undoubtedly be incorporated into a general homotopy theory for topoi.”

The last sentence is now literally true; these theorems are a simple consequence of the “descent” property of spaces, which in fact hold in any $(\infty,1)$-topos. I’ve often wondered if Quillen actually had something like this in mind when he wrote that.