Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: why model categories?

Mike Shulman (Jul 10 2025 at 14:58):

If we're talking about "favorite" models, I have a class of favorites that I suspect ought to encompass enough models for everyone's purposes. Namely, I prefer models whose underlying structure is a diagram of spaces ($\infty$-groupoids) rather than sets. The ur-example is complete Segal spaces. The advantage of models like this is that they are "natively homotopy-invariant". Three concrete manifestations of this are:

1. They tend to arrange into model categories whose weak equivalences between bifibrant objects are "levelwise". Such model categories often have better formal properties.
2. They can be defined internally to any sufficiently structured $(\infty,1)$-category. When working in an arbitrary $(\infty,1)$-category, an arbitrary object generalizes a space, not a set, and such an object may not even have an "underlying set of points" in any sense. So even people who use non-homotopy-invariant models in general (like Lurie using quasicategories) are forced to switch to homotopy-invariant models when internalizing; so why not be consistent and use those models all the time?
3. They can at least potentially be defined in a foundational system like homotopy type theory, whose basic objects are $\infty$-groupoids rather than sets.

Ruby Khondaker (she/her) (Jul 10 2025 at 15:07):

ah i see what you mean - if we had a good notion of "directed space", would that help give a way to talk about higher morphisms being invertible without having to worry about the particular shapes we use too much? so in this case the diagram would be of $(\infty, \infty)$-categories...?

Mike Shulman (Jul 10 2025 at 15:19):

$\infty$-groupoids tend to work much better than any kind of category with noninvertible cells as "raw material" for building things out of. Fundamentally an $\infty$-groupoid just enhances the notion of "equality" in a set to allow things to be equal in more than one way; noninvertible morphisms are a much bigger change.

Mike Shulman (Jul 10 2025 at 15:20):

C.f. e.g. https://mathoverflow.net/q/309515/49

Ruby Khondaker (she/her) (Jul 10 2025 at 15:21):

Ok that makes sense - I was looking into Theta-spaces and I see that they let you model $(\infty, n)$-categories as globular $n$-categories "internal" to spaces? So the space itself only being an $\infty$-groupoid doesn't restrict the level of morphisms that can be invertible, then

Mike Shulman (Jul 10 2025 at 16:22):

Yes, you have a space of objects, a space of 1-morphisms, a space of 2-morphisms, etc.

Ruby Khondaker (she/her) (Jul 10 2025 at 16:53):

So I've been looking more carefully into the definition of Theta spaces, and as far as I can tell, they seem similar to what you describe? They're presheaves of spaces on Joyal's disk category, satisfying:

• An fibrancy condition
• A segal condition
• A completeness condition

I think I understand these best in reverse order, and I have a question about the fibrancy condition, if that's alright.

So, the completeness condition can be nicely stated as the following. Consider the map which "sends a (k-1)-cell to its identity k-cell". Then this induces a weak equivalence between the space of (k-1)-cells, and the space of k-equivalences.

The segal condition I think can be phrased in the following way. Joyal's disk category $\Theta$ seems to essentially be the "category of all globular pasting diagrams". So if we consider the globe category $\mathbb{G}$, then the natural inclusion $\mathbb{G} \to \Theta$ should be dense - this is just the statement that every pasting diagram is a (canonical) pasting together of globes.

Thus, if one has a presheaf on the globe category, you should be able to uniquely extend it to a presheaf on the disk category that sends these colimits to limits - i.e., it's determined by its value on the globe category. As far as I can tell, this is the Segal condition? It seems to tell you that there's not actually a need to define a presheaf on the full disk category - you can simply define it on the globe category, and extend using density.

The injective fibrancy condition is the one I'm currently struggling to get - it's some sort of right lifting property? Is this what's supposed to encode the data of "composition"?

Also I suppose - if the segal condition is going to hold anyway, why consider presheaves on $\Theta$ in the first place? At least to me, it seems simpler to consider presheaves on $\mathbb{G}$, and then extend them to presheaves on $\Theta$ through density. Perhaps I've misunderstood what density gives you...

Mike Shulman (Jul 10 2025 at 16:57):

The Segal condition says that the space of diagrams of a given shape is determined by its spine. This is what gives you composition because a diagram has more boundary than just its spine. The simplest example is the pasting diagram $\cdot \to\cdot \to\cdot$, which is also known as the 2-simplex $\Delta^2$ under the embedding of the simplex category in $\Theta$. The Segal condition says that an element of $X(\Delta^2)$ is determined by two composable morphisms. But $\Delta^2$ also has a third 1-morphism in its boundary, and so you get composition of morphisms by taking two composable morphisms, passing across the Segal condition to a 2-simplex, and then taking the other 1-morphism in its boundary.

Mike Shulman (Jul 10 2025 at 16:59):

The injective fibrancy condition is more of a technical one. From a model-categorical perspective, you need some condition like that to ensure that every homotopy-coherent-natural transformation between your $\Theta$-spaces can be represented by a strictly natural one. From a type-theoretic perspective, it says that the space of 2-simplices (say) is dependent on its boundary, so you can more easily talk about "the space of 2-simplices with such-and-such boundary".

Ruby Khondaker (she/her) (Jul 10 2025 at 17:06):

Oh the injective fibrancy part seems quite interesting to me, then - I think I've seen this sort of phenomenon across higher category theory, actually? It seems to always be the case that the following holds:

• Suppose you have some big pasting diagram of n-cells, and you want to compose it to a single n-cell
• First, you must specify "boundary conditions" - i.e., you must choose a way to compose the boundary of your pasting diagram to the boundary of the n-cell
• Then, given such boundary conditions, there's a unique (read: contractible) space of ways to compose the big pasting diagram into the n-cell.

I guess as a physicist, I view this as some kind of "integrability" condition? That to compose a diagram of shape $\alpha$ into a diagram of shape $\beta$, it suffices to specify boundary conditions, i.e. a choice of composing $\partial \alpha$ to $\partial \beta$. The categorical structure then lets you "integrate" the whole shape $\alpha$ into $\beta$.

It's almost like the pasting diagram is some kind of "combinatorial differential equation", and the condition is that, for specified boundary conditions, there's a unique way to "integrate" this equation.

Is what I said at all coherent? :sweat_smile: . And is there a way to relate it to this injective fibrancy condition?

Mike Shulman (Jul 10 2025 at 17:13):

I suppose that's not an unreasonable analogy as long as you don't take it too far.

Mike Shulman (Jul 10 2025 at 17:13):

I would say that's more about the Segal condition than the injective fibrancy condition, though.

Mike Shulman (Jul 10 2025 at 17:14):

(I'm glad that you've already internalized "unique = contractible" though... (-: )

Mike Shulman (Jul 10 2025 at 17:15):

Injective fibrancy is really just about making certain things strictly equal rather than up to homotopy. Without it, but with the Segal condition, it would still be true that if you specify boundary composites there's a unique way to compose the big pasting diagram compatibly, but the "compatibility" would only be up to homotopy.

Mike Shulman (Jul 10 2025 at 17:16):

(At least, if you formulate the Segal condition correctly involving homotopy pullbacks. In the presence of injective fibrancy you can equivalently formulate it using strict pullbacks, and people usually do.)

Ruby Khondaker (she/her) (Jul 10 2025 at 17:31):

Mike Shulman said:

I suppose that's not an unreasonable analogy as long as you don't take it too far.

Ah ok nice! This is essentially how L’ models composition, I think? Hmm, and why do we need things to be strictly equal - or in other words, why is injective fibrancy helpful?

One nice thing about L’ is that you don’t even have to say contractibility from the start! You can just say “given boundary conditions, there exists a composite”, and then you can prove that the space of such composites must be contractible.

Mike Shulman (Jul 10 2025 at 17:34):

Not contractible in the same sense, since L' is built on sets rather than spaces.

Mike Shulman (Jul 10 2025 at 17:35):

Ruby Khondaker (she/her) said:

and why do we need things to be strictly equal - or in other words, why is injective fibrancy helpful?

Mike Shulman said:

The injective fibrancy condition is more of a technical one. From a model-categorical perspective, you need some condition like that to ensure that every homotopy-coherent-natural transformation between your $\Theta$-spaces can be represented by a strictly natural one. From a type-theoretic perspective, it says that the space of 2-simplices (say) is dependent on its boundary, so you can more easily talk about "the space of 2-simplices with such-and-such boundary".

Ruby Khondaker (she/her) (Jul 10 2025 at 17:35):

Mike Shulman said:

Not contractible in the same sense, since L' is built on sets rather than spaces.

Ah sure, but it’s not too hard to construct the “globular set of all composites”, and then you can prove this is contractible as a globular set, right?

Mike Shulman (Jul 10 2025 at 17:35):

Maybe? I don't know much about L'.

Ruby Khondaker (she/her) (Jul 10 2025 at 17:37):

Ah well - it’s relatively straightforward to say what it means for a globular set to be contractible. “Between any parallel n-cells, there’s a n + 1 cell” is the gist. And then because of the way L’ works, if you fix boundary conditions, then the globular set of composites (where 0-cells are composites of the diagram now) is contractible

Ruby Khondaker (she/her) (Jul 10 2025 at 17:49):

I remember that when I saw it, it seemed kinda similar to the segal space definition? where you assert that the boundary inclusion $\partial \alpha \to \alpha$ induces a map from $X(\alpha) \to X(\partial \alpha)$ with contractible fibers - though I may have misunderstood the similarity..

Mike Shulman (Jul 10 2025 at 18:11):

Kinda similar, certainly!

Mike Shulman (Jul 10 2025 at 18:12):

I think in L' the boundary relative to the "multicategory domain" map $X_1 \to TX_0$ is analogous to the "spine" in a Segal condition. The other parts of the boundary include the "multicategory codomain" $X_1 \to X_0$ that assigns the actual composite.

Ruby Khondaker (she/her) (Jul 10 2025 at 18:23):

Mike Shulman said:

I think in L' the boundary relative to the "multicategory domain" map $X_1 \to TX_0$ is analogous to the "spine" in a Segal condition. The other parts of the boundary include the "multicategory codomain" $X_1 \to X_0$ that assigns the actual composite.

Yes, exactly! Though this is why I prefer when $X_1$ is just a relation - I.e. you just say yes or no when a pasting diagram composed to a cell. Then the multi category axioms are reflexivity and transitivity of this relation, and all you need to get a weak omega-category is to impose an “integrability” condition on the relation. Which says that to compose alpha to beta, it suffices to choose a composite of partial alpha to partial beta!

Fernando Yamauti (Jul 10 2025 at 18:27):

Mike Shulman said:

2. They can be defined internally to any sufficiently structured $(\infty,1)$-category. When working in an arbitrary $(\infty,1)$-category, an arbitrary object generalizes a space, not a set, and such an object may not even have an "underlying set of points" in any sense. So even people who use non-homotopy-invariant models in general (like Lurie using quasicategories) are forced to switch to homotopy-invariant models when internalizing; so why not be consistent and use those models all the time?

What's the problem with internal quasicategories? The completness condition could also be imposed on top of that, and if recall correctly one could state the Segal conditions using Horn fillers and vice-versa.

Shouldn't internal quasicats in anima be equivalent to internal segal cats in anima?

Mike Shulman (Jul 10 2025 at 18:46):

I don't see how you could call that a "quasicategory" any more then. A quasicategory has no completeness condition, and it merely has horn-fillers that exist -- you can state the Segal condition using horns rather than spines but it has to be stated as an equivalence of spaces, not the mere existence of fillers.

Fernando Yamauti (Jul 10 2025 at 19:45):

Mike Shulman said:

you can state the Segal condition using horns rather than spines but it has to be stated as an equivalence of spaces, not the mere existence of fillers.

Sure. I can't recall now a work using internal quasicats, but there are several papers in geometry using internal Kan simplicial sets inside certain higher topoi, and, in those, the lifting conditions are stated also as equivalences internally.

So, in the end, we agree that except for the completeness condition they are the same (internal quasicats and internal Segal cats)? Or perhaps you have something more subtle in mind?

Mike Shulman (Jul 10 2025 at 22:00):

I don't agree that you can call it an "internal quasicategory" any more when you change from the existence of horn-fillers to an equivalence condition. I mean, maybe people do, but I don't think that is justified. An "internal quasicategory" should be something that, when you specialize it to the category of sets or of spaces, yields an ordinary quasicategory; but what you're suggesting does not have that property. When you specialize it to the category of sets, you get the nerves of 1-categories; and when you specialize it to the category of spaces, you get Segal spaces. (And adding the completeness condition makes it gaunt 1-categories and complete Segal spaces, respectively.) So unless you're willing to also call complete Segal spaces "quasicategories", which would destroy the whole point of having separate names for different models of $(\infty,1)$-categories, I don't think you can call those "internal quasicategories".

John Onstead (Jul 10 2025 at 22:37):

Mike Shulman said:

If a model category $C$ is suitably enriched over some other model category $V$ which presents some $(\infty,1)$-category $hV$, then its presented $(\infty,1)$-category $hC$ will be enriched over $hV$. So if you can pick an appropriate $V$ such that an $hV$-enriched $(\infty,1)$-category can be viewed as an $\omega$-category, you can use a $V$-enriched model category $C$ to present an $\omega$-category.

Sorry for interjecting to this discussion, I have a question about the above. What if we picked a particular model of $\omega$ category and defined $hV$ to be the $(\infty,1)$-category of $\omega$ categories of this particular model. Then wouldn't a category enriched in this category also be a viable model for an $\omega$ category? Going by the same logic as saying a $(n+1)$ category is a category enriched in $n\mathrm{Cat}$, and $\omega + 1 = \omega$.

Ruby Khondaker (she/her) (Jul 10 2025 at 22:40):

(first I’ve heard of $\omega + 1 = \omega$)!

John Baez (Jul 10 2025 at 22:46):

People usually define ordinal addition so that

$\omega + 1 > \omega = 1 + \omega$

so maybe John meant to write $1 + \omega = \omega$. But I think the idea is clear: a category weakly enriched in $\omega$-categories should be, at least morally speaking, an $\omega$-category.

Mike Shulman (Jul 10 2025 at 23:21):

John Onstead said:

Then wouldn't a category enriched in this category also be a viable model for an $\omega$ category?

Indeed! Although depending on your model, it would be some work to prove it equivalent to the original model of $\omega$-category you started with.

Mike Shulman (Jul 10 2025 at 23:23):

John Baez said:

People usually define ordinal addition so that

$\omega + 1 > \omega = 1 + \omega$

There's also the natural operations according to which $\omega+1=1+\omega=\omega$. Wikipedia writes the natural sum with $\oplus$, but personally I prefer to only use the symbol $+$ for commutative operations (e.g. the operation in an abelian group, or in this case abelian monoid); so it makes more sense to me for $+$ to mean the natural operation and $\oplus$ to mean the noncommutative one.

Mike Shulman (Jul 10 2025 at 23:24):

Wikipedia's definition of the "natural sum" makes it look anything but natural, but in fact it's just the addition of the [[surreal numbers]] specialized to ordinals.

Oscar Cunningham (Jul 11 2025 at 06:33):

Mike Shulman said:

There's also the natural operations according to which $\omega+1=1+\omega=\omega$.

I think there's a typo here. In the natural operations we do have that $\omega+1=1+\omega$, but they're both greater than $\omega$.

Mike Shulman (Jul 11 2025 at 14:32):

Oops! Fixed, thanks.

Mike Shulman (Jul 11 2025 at 15:10):

Although actually, I don't know any more why I mentioned the natural operations here, since it does seem to be the "unnatural" ones that are relevant for indexing categories. Saying that an $\omega$-category-enriched category is an $\omega$-category would indeed correspond to $1+\omega =\omega$ as John said, and on the other hand there is also a different notion of $(\omega+1)$-category that has $\omega$-cells.

Ruby Khondaker (she/her) (Jul 11 2025 at 15:17):

$\omega$-cells sound terrifying - what would their source and target even be?? Though I guess, perhaps when $\omega$-category theory is mainstream, we'll have people pursuing ordinal category theory...

I for one am looking forward to $\varepsilon_0$-category theory~

Mike Shulman (Jul 11 2025 at 16:27):

The boundary of a $\kappa$-cell, for any ordinal $\kappa$, is a sequence $\{(x_\lambda,y_\lambda)\}_{\lambda<\kappa}$ of pairs of $\lambda$-cells for all $\lambda<\kappa$, such that both $x_\lambda$ and $y_\lambda$ have boundary $\{(x_\mu,y_\mu)\}_{\mu<\lambda}$. You can check that that gives the usual thing for $\kappa$ finite.

Mike Shulman (Jul 11 2025 at 16:30):

The only place I know of that this appears in the literature is sort of "by accident". You can generalize the [[single-sorted definition of category]] to a definition of strict $n$-category where the single set of objects is the $n$-cells, with $k$-cells for $k<n$ represented by their identity $n$-cells. You can then try to do the same for strict $\omega$-categories, but if you then neglect to add a condition along the lines of "all cells have finite dimension" what you actually get is a definition of strict $(\omega+1)$-categories.

John Onstead (Jul 11 2025 at 18:10):

I certainly made a mistake when I said $\omega + 1 = \omega$, maybe I should have used the $\infty$ symbol instead? But either way it's led to some interesting discussion!

John Onstead (Jul 11 2025 at 18:11):

Though speaking of source and target... This might have been asked already here, but I was wondering what model of infinity category is most naturally associated with the idea of a "path monad". For instance, a 1-category has an underlying directed graph, a presheaf on the category with two objects (representing 0-cells and 1-cells) and two morphisms from one to the other (representing source and target). The path monad works by freely adding in all possible compositions, so the resulting algebras for the monad are 1-categories themselves. In the 2-categorical case, we have a path monad on presheaves on the category with 3 objects with source and target morphisms from the first to second and second to third. And so on with higher categories, with more and more objects and source/target morphisms from one to the next.

John Onstead (Jul 11 2025 at 18:11):

This makes me wonder: what if you take presheaves on a category with infinite objects organized into an infinite chain, where each has a source and target morphism from the predecessor and to its successor? Is there a valid path monad on this presheaf category and if so which kind of infinity category might you get?

Mike Shulman (Jul 11 2025 at 18:12):

Yes, that sounds like the Batanin-Leinster "globular operad" approach to $\omega$-categories.

Ruby Khondaker (she/her) (Jul 11 2025 at 18:13):

John Onstead said:

This makes me wonder: what if you take presheaves on a category with infinite objects organized into an infinite chain, where each has a source and target morphism from the predecessor and to its successor? Is there a valid path monad on this presheaf category and if so which kind of infinity category might you get?

This sounds like globular sets? In which case you can look at Leinster's operad, whose algebras give you weak $\omega$-categories. He covers it in his book Higher Operads, Higher Categories

Ruby Khondaker (she/her) (Jul 11 2025 at 18:13):

Ah, I got sniped!

Chris Grossack (she/they) (Jul 13 2025 at 12:47):

I mean, I don't really know what Quillen had in mind. But I mean what I said -- model categories are a useful tool for doing concrete computations in oo-categories because they let you find "nice" objects with which a 1-categorical computation in the model category (where you don't have to keep track of higher homotopies) agrees with the correct oo-categorical computation in the oo-category the model category presents (where you do have to keep track of higher homotopies)

fosco (Jul 13 2025 at 14:35):

Yes, I'm just saying that this is a ex-post justification, it's like saying that "Fourier theory is about certain decompositions into irreducible representations". It's not wrong, no one would debate that. But it's not what Fourier did, or what he had in mind, it's a subsequent reinterpretation of what the theory is "really" about.

fosco (Jul 13 2025 at 14:37):

The progress between Whitehead speech at the ICM of 1950 and Quillen definition, with all its dead ends and attempts at a general synthetic homotopy theory, is very well documented by the way.

Mike Shulman (Jul 13 2025 at 15:19):

An ex-post justification is still perfectly valid as an explanation of the subject for a modern newcomer.

fosco (Jul 13 2025 at 17:02):

This is not a hill I want to die on... I just wanted to say what I said

fosco (Jul 13 2025 at 17:04):

I tend to dislike ex-post justifications, I prefer to be told how things went

Mike Shulman (Jul 13 2025 at 17:10):

Well, if you want to contribute some historical background to the conversation I'm sure it would also be welcome.

Chris Grossack (she/they) (Jul 13 2025 at 20:42):

I know I would be interested!

Mike Shulman (Jul 13 2025 at 21:58):

I have not myself done actual research into the history, but what I've picked up from older mathematicians is that algebraic topology started out, long before anyone even knew what a 1-category was let alone an $\infty$-category, by trying to define and compute invariants of topological spaces. Then gradually people noticed that nearly all the invariants anyone had come up with were not only invariants of homeomorphism but also of homotopy equivalence, and began gradually to study the notion of "topological space up to homotopy equivalence". Fibrations and cofibrations were invented as tools for doing this and for computing these invariants.

As Mac Lane famously said, categories were defined in order to be able to define functors, and functors were defined in order to be able to define natural transformations -- specifically, I believe, natural transformations between homotopy invariants of topological spaces such as the connecting morphisms in long exact sequences for homology and cohomology. They gradually became more and more explicit in algebraic topology, and eventually Quillen, like a good category theorist, observed what people were already doing with fibrations and cofibrations in concrete situations and wrote down some abstract axioms describing it.

Notification Bot (Jul 14 2025 at 07:57):

John Baez has marked this topic as unresolved.

John Baez (Jul 14 2025 at 08:15):

I started struggling to read Quillen's Homotopical Algebra well before $(\infty,1)$-categories were born, and I remember Peter May talking about how he was slow to adopt model categories but eventually became a convert. So the historical reasons for model categories are still fairly clear to me.

As far as I can tell, they became important in algebraic topology for reasons similar to why categories were invented. Mac Lane and Eilenberg were struggling with a diversity of approaches to defining the ordinary cohomology of topological spaces that seemed "naturally isomorphic": before there was a definition of natural isomorphism, this was just a hand-wavy feeling, so they couldn't prove a theorem about one approach using another approach. As Mac Lane later wrote,

As Freyd has observed, "category" has been defined in order to define "functor", and "functor" has been defined in order to define "natural transformation".

In fact just defining "natural isomorphism" would already have been worth a lot.

Later, when algebraic topology had matured, it was clear that everything you could do in homotopy theory with topological spaces you could do with simplicial sets. The subtlety was that there are things you can do with one that you can't do with the other - the categories aren't equivalent, after all! - but these things don't count as "in homotopy theory". So Quillen wanted to know in what subtle sense these categories were "the same for the purposes of homotopy theory", which required figuring out the essential features of "homotopy theory". He was led to model categories and what we now call "Quillen equivalence" of model categories.

In parallel he was thinking hard about chain complexes and honological algebra and its use of projective and injective resolutions. He knew that the category of chain complexes had a lot of similar structures to that of topological spaces and simplicial sets - indeed this was generally known in a hand-wavy way - but he wanted to make the analogy precise. I think projective and injective resolutions helped lead him to the concepts of cofibrant and fibrant replacement.

He thought he was extending homological algebra to a broader framework, hence the title Homotopical Algebra.

Mike Shulman (Jul 14 2025 at 15:15):

One thing I'm not entirely clear on is when, and to who, it first became clear that "abstract homotopy theory" was closely related to "$(\infty,1)$-category theory". I gather that Grothendieck was (one of?) the first to propose that classical homotopy types should be equivalent to some notion of $\infty$-groupoid, but I don't know when that was generalized to abstract homotopy theory and $(\infty,1)$-categories.

John Baez (Jul 14 2025 at 21:45):

That's a great question. I associate the idea "model categories present $(\infty,1)$-categories" with Jacob Lurie. But before him (as you know) a bunch of people were noticing that you can sometimes change the fibrations and cofibrations in a model category while keeping the weak equivalences the same, thus describing the same homotopy theory in some sense. This prompted various weakenings of the model category axioms, all the way down to the concept of [[category with weak equivalences]]. This goes back a long ways; the nLab says

The terminology and specific definition of "category with weak equivalences" appears in:

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan and Jeffrey H. Smith, §2.3 of: Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113, American Mathematical Society (2004) [doi:10.1090/surv/113, pdf]

though the basic idea goes back, at least, to:

• William Dwyer, Daniel Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267-284. [doi:10.1016/0022-4049(80)90049-3]

Mike Shulman (Jul 14 2025 at 22:17):

I would have put it earlier than Lurie; I thought I remembered it being fairly current even when I was just starting as a graduate student circa 2004. For instance, isn't a slogan like "model categories present $(\infty,1)$-categories" assumed by the title "A model for the homotopy theory of homotopy theories" of Rezk's 2001 paper on the model category of complete Segal spaces?

Kevin Carlson (Jul 14 2025 at 22:27):

I don't see Rezk thinking about $(\infty,1)$-categories in that paper! As he says in his intro and John in his comment, Dwyer and Kan were already thinking about a "homotopy theory of homotopy theories", since you can define a relative category of relative categories whose weak equivalences are the maps inducing a DK-equivalence on simplicial localizations, just not in terms of a homotopy theory being an $(\infty,1)$-category. You can find Joyal saying "Quasi-categories are special cases of weak $\omega$-categories: exactly those having only invertible cells in dimensions $>1$" in https://www.sciencedirect.com/science/article/pii/S0022404902001354, though. He mentions this rather casually, so perhaps Cordier and Porter already had (essentially?) the same idea in their papers on quasicategories in the '90s.

Mike Shulman (Jul 14 2025 at 22:32):

Rezk doesn't use the phrase "$(\infty,1)$-category", but how can a model category of complete Segal spaces be "a homotopy theory of homotopy theories" unless complete Segal spaces (which we now, at least, can recognize as a model for $(\infty,1)$-categories) are identified with "homotopy theories" and also model categories give rise to homotopy theories?

Mike Shulman (Jul 14 2025 at 22:33):

He even explains how a model category gives rise to a complete Segal space.

Kevin Carlson (Jul 14 2025 at 22:57):

I would agree that Rezk clearly understands that model categories present homotopy theories. I think there's a separate identification to consider between homotopy theories and $(\infty,1)$-categories.

Mike Shulman (Jul 14 2025 at 23:51):

Would you also agree that since he identifies "homotopy theories" with complete Segal spaces, what he means by "homotopy theories" is the same as what we now call "$(\infty,1)$-categories"?

Kevin Carlson (Jul 15 2025 at 00:14):

I mostly think yes, but then Dwyer and Kan also meant "the same thing" as our $(\infty,1)$ categories by their homotopy theory of relative categories, no? Insofar as they were talking about the same $(\infty,1)$-category as we are now?

Mike Shulman (Jul 15 2025 at 00:16):

Was there a homotopy theory of relative categories before Barwick-Kan 2012?

Mike Shulman (Jul 15 2025 at 00:17):

Prior to that my impression was that relative categories (then known as categories with weak equivalences or homotopical categories) were mainly regarded as presentations of homotopy theories, like model categories.

Kevin Carlson (Jul 15 2025 at 00:27):

Hm, I was assuming that in writing a paper that defines weak equivalences of simplicially enriched categories and the homotopy coherent nerve of a relative category, DK must have noted somewhere that this gives a relative category of relative categories, or a homotopy theory of homotopy theories. But I can't find a clear such point in there. Maybe they didn't quite get there!

Mike Shulman (Jul 15 2025 at 00:35):

I have a pretty strong memory that at the 2004 IMA meeting on $n$-categories, when people were discussing how to go about comparing different definitions of $n$-category, Julie Bergner's thesis establishing Quillen equivalences between complete Segal spaces, Segal categories, and simplicial categories was pointed out as an example of such a comparison in the case of $(\infty,1)$-categories. I suppose it's possible that I'm back-projecting my later understanding onto my past self, but I don't think so.

Mike Shulman (Jul 15 2025 at 00:58):

I was surprised, just now, on looking back at Rezk and Bergner's papers, that they don't appear to use the phrase $(\infty,1)$-category yet. Perhaps at that point it was in informal circulation, but not thought sufficiently respectable to appear in a published paper (at least, an algebraic topology paper)?

John Baez (Jul 15 2025 at 10:05):

Quite possibly! It kind of assumes at least a handwavy understanding of general $\infty$-categories, so one can say "an $\infty$-category where all $n$-morphisms with $n \ge 1$ are equivalences". Some people (like me!) are more willing than others to say such things before it's 100% clear what they mean.

So what's the first known appearance of the term $(\infty,1)$-category, or $(\infty,k)$-category, or $(n,k)$-category?

David Corfield (Jul 15 2025 at 11:01):

John Baez said:

So what's the first known appearance of the term $(\infty,1)$-category, or $(\infty,k)$-category, or $(n,k)$-category?

Wouldn't surprise me if it was in one of your papers. I seem to recall people were speaking of $\infty$-categories and $n$ Café people always wanted to correct to $(\infty, 1)$-categories.

When's the first appearance of $\infty$-category in the sense of $(\infty, 1)$? I see Toen and Vezzosi writing it in 2001.

Mike Shulman (Jul 15 2025 at 16:22):

There might even be one or two early papers that say $(1,\infty)$-category instead.

Mike Shulman (Jul 15 2025 at 16:33):

The 2010 book Towards Higher Categories which is not a proceedings of the 2004 IMA conference seems to agree with my memory.
From the preface, presumably written by its editors John C. Baez and J. Peter May:

A smaller related theme of the conference was that there should be a "baby" comparison project, for which model category theory would in fact be sufficient. Precisely, the idea was that there should be a web of Quillen equivalences among the various notions of $(\infty,1)$-category. These include topologically or simplicially enriched categories, Segal categories, complete Segal spaces, and quasi-categories. In the years since the conference, this comparison project has been largely completed by Bergner and Joyal and Tierney.

John Baez (Jul 16 2025 at 08:11):

At that time we knew about Julie Bergner's 2006 paper A survey of $(\infty, 1)$-categories.

David Corfield (Jul 16 2025 at 11:23):

She refers to Toen's thesis which contains a 2001 letter to Peter May. No sign of the $(n,k)$ notation, but there are things like:

image.png

John Baez (Jul 16 2025 at 12:00):

David Corfield said:

John Baez said:

So what's the first known appearance of the term $(\infty,1)$-category, or $(\infty,k)$-category, or $(n,k)$-category?

Wouldn't surprise me if it was in one of your papers.

It would surprise me. I was thinking a lot about $k$-tuply monoidal $n$-categories, and $k$-tuply groupal $n$-groupoids, but not much about $(\infty,n)$-categories. I was certainly aware of Joyal's quasicategories, and Segal categories, and simplicially enriched categories, and eventually the work of Julia Bergner and others comparing these. But I don't remember ever trying to explain this work.

I generally regarded the move toward $(\infty,1)$-categories as "the invasion of the experts" - that is, experts on homotopy theory. When the experts took over, I realized my day of working on higher categories was done and I needed to do something new. Then I followed my wife to Singapore, got a job at the Centre for Quantum Technologies, and started working on climate change and applied category theory.

So, while I could have explained $(\infty,1)$-categories at some point, and the idea of $(\infty,n)$-categories, I don't remember ever doing it. Those were never ideas I was pushing.

Urs, on the other hand, took to them!