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Stream: learning: questions

Topic: ✔ why model categories?

Ruby Khondaker (she/her) (Jul 09 2025 at 20:03):

I keep seeing them crop up in higher category theory, and I only barely understand their definition - I don’t really have a good conceptual or “intuitive” reason for why they’re good

Chris Grossack (she/they) (Jul 09 2025 at 21:21):

To oversimplify dramatically, model categories are a computational tool. When working with oo-categories, there's many concepts that are only defined "up to a sequence of higher homotopies". This makes it almost impossible to do concrete computations (eg. limits/colimits, but already computing the space Hom(X,Y) can be difficult), since it's cumbersome to have to handle this sequence of higher homotopies by hand

Ruby Khondaker (she/her) (Jul 09 2025 at 21:21):

oooh that sounds pretty good - i'm a physicist, i love computational tools!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:22):

i have heard that they "present" an infinity category in some appropriate sense

Ruby Khondaker (she/her) (Jul 09 2025 at 21:23):

is this in a similar way to how you can "present" a group by generators and relations?

Chris Grossack (she/they) (Jul 09 2025 at 21:23):

A model category is a 1-category which gives a particular model for this oo-category, in the sense that it's a particular (non-canonical) presentation, and in which computations are easier. This is essentially because they handle the sequence of higher homotopies for you, and turn your computation into something 1-categorical

Ruby Khondaker (she/her) (Jul 09 2025 at 21:23):

oh, that's awesome!

Chris Grossack (she/they) (Jul 09 2025 at 21:24):

Yeah, they're SUPER useful!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:24):

:D

Ruby Khondaker (she/her) (Jul 09 2025 at 21:24):

how do they manage to package everything into 1-categorical concepts?

Ruby Khondaker (she/her) (Jul 09 2025 at 21:24):

presumably not every $\infty$-category admits a model category presentation?

Ruby Khondaker (she/her) (Jul 09 2025 at 21:25):

as in you can turn a model category into an $\infty$-category but not the other way around

Chris Grossack (she/they) (Jul 09 2025 at 21:26):

So, for example, say you want to do a computation with the oo-category of spaces. You can "model" spaces as simplicial sets, or by topological spaces. You get two different model categories (with an adjunction between them, telling you how to translate back and forth between the models), and you can compute in either one, depending on which is easier.

Ruby Khondaker (she/her) (Jul 09 2025 at 21:26):

ah hm ok ok

Ruby Khondaker (she/her) (Jul 09 2025 at 21:26):

so i have heard of, say, quasicategories, as a particular subcategory of simplicial sets

Ruby Khondaker (she/her) (Jul 09 2025 at 21:27):

simplicial sets on their own are just a presheaf category, so a 1-category, right?

Ruby Khondaker (she/her) (Jul 09 2025 at 21:27):

objects are simplicial sets, morphisms are natural transformations between these

Ruby Khondaker (she/her) (Jul 09 2025 at 21:27):

if you wanted to make this into an $(\infty, 1)$ category you'd need modifications and higher transfors

Ruby Khondaker (she/her) (Jul 09 2025 at 21:28):

but, you're saying that you don't actually need to do this? in fact, you can get away just with morphisms of simplicial sets?

Chris Grossack (she/they) (Jul 09 2025 at 21:28):

Simplicial sets are extremely flexible, and by choosing different model structures they can model different things. For instance, with one model structure simplicial sets model spaces. With another model structure, simplicial sets model quasi-categories (read: oo-categories). Concretely what this means is that you can reduce many oo-categorical questions about spaces or oo-categories to 1-categorical questions about simplicial sets, by using the appropriate model structure. But now you're doing combinatorics, so you have a chance at actually doing the computation!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:29):

awesome!!!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:29):

so a model structure on a 1-category is, roughly, a way of "automatically generating all higher cells"?

Ruby Khondaker (she/her) (Jul 09 2025 at 21:29):

but in a way that can be described purely in 1-categorical terms

Ruby Khondaker (she/her) (Jul 09 2025 at 21:30):

and different model structures correspond to different choices of higher cells

Chris Grossack (she/they) (Jul 09 2025 at 21:30):

Kind of! It's maybe better to think of it as a way to cleverly choose a representative object without many higher cells, so that the oo-computation ends up just being a 1-computation after all

Ruby Khondaker (she/her) (Jul 09 2025 at 21:31):

hm, how do you mean?

Chris Grossack (she/they) (Jul 09 2025 at 21:32):

All the homotopy theory/higher cells/etc is stored in the weak equivalences. The way you often get oo-categories is by taking a 1-category and freely inverting some set of arrows that morally should be isomorphisms. The classic examples are topological spaces (where we want to invert the homotopy equivalences) and chain complexes (where we want to invert the quasi-isomorphisms)

Ruby Khondaker (she/her) (Jul 09 2025 at 21:32):

hm i have roughly heard this before but didn't quite understand it

Ruby Khondaker (she/her) (Jul 09 2025 at 21:32):

if you freely invert a set of arrows, where you do get the higher cells from?

Ruby Khondaker (she/her) (Jul 09 2025 at 21:33):

like e.g. what would a 2-cell be in this context

Chris Grossack (she/they) (Jul 09 2025 at 21:33):

Now, of course, when you invert the weak equivalences, things that used to not be isomorphic become isomorphic (that's the whole point). So when you do a computation in the localized category (really an oo-category) any one of these objects is as good as any other. The model structure tells you how to choose a good object, on which the obvious 1-categorical computation will actually agree with the more subtle oo-categorical computation

Ruby Khondaker (she/her) (Jul 09 2025 at 21:33):

ohhhhh

Ruby Khondaker (she/her) (Jul 09 2025 at 21:34):

so like, for $\mathbf{Top}$

Ruby Khondaker (she/her) (Jul 09 2025 at 21:34):

$\mathbb{R}^n$ for distinct $n$ are not homeomorphic spaces

Ruby Khondaker (she/her) (Jul 09 2025 at 21:34):

but if you say that homotopy equivalences really should be isomorphisms, then they are

Chris Grossack (she/they) (Jul 09 2025 at 21:35):

exactly!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:35):

and so a model category might say "hey, it might be useful to use $*$ for this problem, and $\mathbb{R}^2$ for this one, and $\mathbb{R}^{420}$ for this one"

Ruby Khondaker (she/her) (Jul 09 2025 at 21:35):

Chris Grossack (she/they) said:

exactly!

:D

Ruby Khondaker (she/her) (Jul 09 2025 at 21:35):

so i guess - just having the weak equivalences isn't enough to tell you what objects are "good"

Ruby Khondaker (she/her) (Jul 09 2025 at 21:36):

is that where the fibrations and cofibrations come in?

Chris Grossack (she/they) (Jul 09 2025 at 21:36):

Also, a few years ago when I was first learning this, I wrote up some blog posts about model categories and whatnot. I haven't reread them recently, but I don't think anything I said was too off-base, so you might find them helpful: https://grossack.site/tags/homotopy-theories/

Ruby Khondaker (she/her) (Jul 09 2025 at 21:36):

i'll check them out, thank you so much!! you've already been really helpful :)

Chris Grossack (she/they) (Jul 09 2025 at 21:37):

Iirc in pt 3 I give an example (that I think I learned from one of Carlos Simpson's lectures?) of how inverting arrows can create higher homotopies, which you might find instructive

Ruby Khondaker (she/her) (Jul 09 2025 at 21:37):

thanks for the heads-up :>

Ruby Khondaker (she/her) (Jul 09 2025 at 21:38):

it's interesting because when i think "homotopy" i think "continuous path between two things"

Chris Grossack (she/they) (Jul 09 2025 at 21:38):

Ruby Khondaker (she/her) said:

and so a model category might say "hey, it might be useful to use $*$ for this problem, and $\mathbb{R}^2$ for this one, and $\mathbb{R}^{420}$ for this one"

Yeah, but a more accurate example would be "instead of using a module, you should use its projective resolution" or, "instead of using a crummy topological space, you should use a weakly equivalent CW complex"

Ruby Khondaker (she/her) (Jul 09 2025 at 21:38):

but apparently "homotopy" really means "these things should be isomorphic even though they're not"?

Ruby Khondaker (she/her) (Jul 09 2025 at 21:39):

Ruby Khondaker (she/her) said:

it's interesting because when i think "homotopy" i think "continuous path between two things"

i guess this is really saying something about "these two morphisms should be the same, even though they're not"

Ruby Khondaker (she/her) (Jul 09 2025 at 21:40):

so i guess homotopy naturally lives at the level of 1-cells in my head, not 0-cells

Chris Grossack (she/they) (Jul 09 2025 at 21:41):

Homotopies can live in all dimensions (I guess maybe $\geq 1$). So if you have two arrows $f,g : X \to Y$ you can have a homotopy $H : f \sim g$. But maybe there's a second homotopy $K : f \sim g$ that also works. You can ask for a higher homotopy $\Theta : H \sim K$. But you might have two of these, and ask for a higher higher homotopy between those, and so on.

Ruby Khondaker (she/her) (Jul 09 2025 at 21:42):

yes yes of course, but what i'm going off is this statement in your blog post:

Ruby Khondaker (she/her) (Jul 09 2025 at 21:42):

image.png

Chris Grossack (she/they) (Jul 09 2025 at 21:42):

oh no, hopefully something I said 3 years ago isn't going to come back to bite me, haha. Let's see

Ruby Khondaker (she/her) (Jul 09 2025 at 21:42):

to me this sounds like you're saying "these two 0-cells should be equivalent, even though they might not be at the moment"

Ruby Khondaker (she/her) (Jul 09 2025 at 21:43):

but for me, homotopy only makes sense for 1-cells and higher?

Chris Grossack (she/they) (Jul 09 2025 at 21:43):

Ah, sure. Yeah that is what I'm saying.

Ruby Khondaker (she/her) (Jul 09 2025 at 21:43):

i think that's where my confusion lies at the moment

Ruby Khondaker (she/her) (Jul 09 2025 at 21:43):

i understand homotopy in the "these two 1-cells should really be identified" (or higher), but not really "this particular 1-cell should really be an iso"

Chris Grossack (she/they) (Jul 09 2025 at 21:44):

Ah, good. This is a perfect time to introduce part of the model category structure, then, since I think it will explain the confusion

Ruby Khondaker (she/her) (Jul 09 2025 at 21:44):

so i guess i need to figure out why these notions are the same idea

Ruby Khondaker (she/her) (Jul 09 2025 at 21:44):

oh!

Chris Grossack (she/they) (Jul 09 2025 at 21:45):

Say you have a map $f : X \to Y$ of topological spaces. Then we say $f$ is a weak homotopy equivalence if it induces isomorphisms $f_* : \pi_n X \overset{\sim}{\to} \pi_n Y$ on all homotopy groups. If we're doing homotopy theory, it makes sense that we might want to consider such an $f$ "as good as an isomorphism".

Ruby Khondaker (she/her) (Jul 09 2025 at 21:46):

right right

Ruby Khondaker (she/her) (Jul 09 2025 at 21:46):

so if you "probe" $f$ by maps out of the $n$-spheres

Ruby Khondaker (she/her) (Jul 09 2025 at 21:46):

then they can't distinguish $f$?

Ruby Khondaker (she/her) (Jul 09 2025 at 21:46):

so long as you identify homotopic maps out of the $n$-spheres

Ruby Khondaker (she/her) (Jul 09 2025 at 21:47):

or sorry, they can't distinguish between $X$ and $Y$

Chris Grossack (she/they) (Jul 09 2025 at 21:49):

But there's another (stronger) thing you can ask for, which is just a homotopy equivalence. This is a pair of maps $f : X \to Y$ and $g : Y \to X$ so that $fg$ and $gf$ are homotopic to their respective identity maps. It's a sad fact of life (reflected in the names) that not every weak homotopy equivalence $f : X \to Y$ is half of a homotopy equivalence

Chris Grossack (she/they) (Jul 09 2025 at 21:51):

What you would LIKE to say is that $\text{Hom}_\text{Ho}(X,Y)$ in the homotopy category should just be $\text{Hom}(X,Y) \big / \sim$ -- the set of arrows in the category we started with, quotiented out by homotopy equivalence. If you did this, then all of our homotopy equivalences would become isomorphisms, and things would be nice and easy to compute with (since we can just work with the arrows in the category we started with, remembering that they're actually equivalence classes now)

Ruby Khondaker (she/her) (Jul 09 2025 at 21:52):

Yeah, that’s what I’d expect!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:53):

And I guess it makes more intuitive sense - now the homotopies only compare 1-cells

Ruby Khondaker (she/her) (Jul 09 2025 at 21:53):

But I’m assuming there’s a catch

Chris Grossack (she/they) (Jul 09 2025 at 21:54):

Unfortunately, this doesn't work! Basically because weak homotopy equivalences don't need to have homotopy inverses. Indeed, if we invert a homotopy equivalence $f : X \to Y$ we already have a good candidate -- its quasi-inverse $g : Y \to X$. This means we can see it at the level of our old homsets. But if $f$ is merely a weak homotopy equivalence, then we really have to add a genuinely new arrow, and this makes our homsets bigger in the homotopy category, not smaller!

Chris Grossack (she/they) (Jul 09 2025 at 21:55):

Indeed you can come up with 1-categories and weak equivalences so that $\text{Hom}(X,Y)$ starts out as a set, but after passing to the homotopy category it becomes a proper class!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:55):

Ah ok so

Ruby Khondaker (she/her) (Jul 09 2025 at 21:56):

If I take homotopy for now to mean “some way of identifying 1-cells”

Ruby Khondaker (she/her) (Jul 09 2025 at 21:56):

Then you can have “strong” homotopy equivalences, which are arrows with a homotopy inverse

Chris Grossack (she/they) (Jul 09 2025 at 21:56):

This seems like a tricky situation, but thankfully Whitehead's Theorem comes to save the day:

Every weak homotopy equivalence between CW complexes is a homotopy equivalence!

Ruby Khondaker (she/her) (Jul 09 2025 at 21:56):

But often it’s the case that you want more than these - “weak homotopy equivalences”

Ruby Khondaker (she/her) (Jul 09 2025 at 21:57):

Which don’t have a homotopy inverse at the level of spaces, but do at the level of homotopy groups, which is really what you care about

Ruby Khondaker (she/her) (Jul 09 2025 at 21:57):

Inverting these then has to add arrows, so you get a bigger homset

Ruby Khondaker (she/her) (Jul 09 2025 at 21:57):

Is that about right…?

Chris Grossack (she/they) (Jul 09 2025 at 21:57):

So this tells you that as long as X and Y are CW complexes, it will be the case that $\text{Hom}_\text{Ho}(X,Y) = \text{Hom}(X,Y) \big / \sim$, which is much easier to compute with!

Chris Grossack (she/they) (Jul 09 2025 at 21:59):

Then one thing the model structure buys you is a good way to take a general space $X$ and replace it by a weakly homotopy equivalent CW-complex $\widetilde{X}$. Then we can essentially define $\text{Hom}_\text{Ho}(X,Y)$ to be $\text{Hom}_\text{Ho}(\widetilde{X},Y) = \text{Hom}(\widetilde{X},Y) \big / \sim$

Ruby Khondaker (she/her) (Jul 09 2025 at 22:00):

oh you don't need to replace $Y$?

Chris Grossack (she/they) (Jul 09 2025 at 22:02):

It depends on your exact model structure. In general you need $X$ to be "cofibrant" and $Y$ to be "fibrant" to make things work out nicely. It turns out that in the classical model structure on topological spaces every object is fibrant, so it suffices to just cofibrantly replace $X$. But in general you might need to replace one or the other or both.

Ruby Khondaker (she/her) (Jul 09 2025 at 22:03):

ohhh ok i've heard of this!

Ruby Khondaker (she/her) (Jul 09 2025 at 22:03):

so hm, lemme check if this is accurate

Ruby Khondaker (she/her) (Jul 09 2025 at 22:03):

suppose $X$ and $Y$ are objects of your model category

Ruby Khondaker (she/her) (Jul 09 2025 at 22:04):

and you want to compute $\text{Hom}_{\text{Ho} }(X, Y)$ in the localised category

Ruby Khondaker (she/her) (Jul 09 2025 at 22:04):

unfortunately just taking equivalence classes doesn't work :(

Ruby Khondaker (she/her) (Jul 09 2025 at 22:04):

but! if you replace $X$ with a cofibrant object $\hat X$, and $Y$ with a fibrant object $\hat Y$

Ruby Khondaker (she/her) (Jul 09 2025 at 22:04):

(that they're weakly equivalent to)

Ruby Khondaker (she/her) (Jul 09 2025 at 22:05):

then $\text{Hom}_\text{Ho}(X, Y) = \text{Hom}(\hat X, \hat Y) / \sim$

Chris Grossack (she/they) (Jul 09 2025 at 22:07):

Exactly! And, moreover, this is $\pi_0$ of the space of maps in the $\infty$-category version of the localization. So you can already see how the model structure is helping us get our hands on the $\infty$-category, even in this easy example.

Ruby Khondaker (she/her) (Jul 09 2025 at 22:07):

oh, wonderful!

Ruby Khondaker (she/her) (Jul 09 2025 at 22:07):

so i'm assuming you can additionally recover 2-cells from $\hat X$ and $\hat Y$?

Ruby Khondaker (she/her) (Jul 09 2025 at 22:08):

and then if you want to compose maps, you want the "middle" object to be bifibrant

Ruby Khondaker (she/her) (Jul 09 2025 at 22:08):

so that you can map into and out of it nicely

Chris Grossack (she/they) (Jul 09 2025 at 22:08):

yup! The trick to getting the 2-cells is to just not quotient out by homotopy. You remember the homotopies $H : f \to g$ as higher data

Ruby Khondaker (she/her) (Jul 09 2025 at 22:09):

mhm mhm - i'm familiar with how this is done in the $\infty$-categorical case, but not yet in the model category case

Ruby Khondaker (she/her) (Jul 09 2025 at 22:11):

oh, i just got to the hammock construction!

Chris Grossack (she/they) (Jul 09 2025 at 22:11):

In the nicest cases, your model category is

(1) enriched in simplicial sets (so that $\text{Hom}(X,Y)$ is always a simplicial set
and
(2) $\text{Hom}(X,Y)$ is a kan-complex (read: bifibrant as a simplicial set) whenever $X$ and $$Y$ are bifibrant (or more generally whenever $X$ is cofibrant and $$Y$ is fibrant)

Chris Grossack (she/they) (Jul 09 2025 at 22:11):

But in general you do this hammock construction that you found, haha.

Ruby Khondaker (she/her) (Jul 09 2025 at 22:12):

hm so this is simplicial localization

Ruby Khondaker (she/her) (Jul 09 2025 at 22:12):

is there a version for cubical localization?

Chris Grossack (she/they) (Jul 09 2025 at 22:12):

(iirc the hammock construction is actually regarded as not the best approach. You probably want "bousfield localization", but I didn't know anything about that when I was writing these posts)

Ruby Khondaker (she/her) (Jul 09 2025 at 22:12):

oh, i've heard of that! i think mike shulman mentioned it in another channel

Chris Grossack (she/they) (Jul 09 2025 at 22:13):

What do you mean by "cubical" localizations? Do you want to model your mapping spaces by cubes instead of simplicies?

Ruby Khondaker (she/her) (Jul 09 2025 at 22:13):

yeah!

Ruby Khondaker (she/her) (Jul 09 2025 at 22:13):

i just prefer cubical sets for a few things

Chris Grossack (she/they) (Jul 09 2025 at 22:15):

Yeah, totally. There's a model structure on cubical sets which presents the $\infty$-category of spaces. So you could probably just as well work with a 1-category enriched in cubical sets with the property that $\text{Hom}(X,Y)$ is bifibrant (as a cubical set) whenever $X$ is cofibrant and $Y$ is fibrant

Ruby Khondaker (she/her) (Jul 09 2025 at 22:15):

:D

Chris Grossack (she/they) (Jul 09 2025 at 22:15):

I haven't thought much about these things, but especially with cubical agda I know a lot of people have thought a lot about it

Ruby Khondaker (she/her) (Jul 09 2025 at 22:15):

the reason why cubical sets are nice is cause they allow for tensor products more easily

Ruby Khondaker (she/her) (Jul 09 2025 at 22:15):

oh is that cubical type theory stuff?

Ruby Khondaker (she/her) (Jul 09 2025 at 22:16):

i think one of my friends is into that

Ruby Khondaker (she/her) (Jul 09 2025 at 22:17):

Also I think Cubical models of $(\infty, 1)$ categories is nice for this :)

Chris Grossack (she/they) (Jul 09 2025 at 22:20):

Cool! Yeah, iirc there were some problems relating the precise semantics of cubical agda to the usual $\infty$-category of spaces? But I'm not sure what the state of the art is on that

Ruby Khondaker (she/her) (Jul 09 2025 at 22:21):

For reference I'm just trying to understand the $(\infty, \infty)$ category of weak $(\infty, \infty)$ categories

Chris Grossack (she/they) (Jul 09 2025 at 22:22):

Anyways, I have to go to bed, but hopefully this was helpful! Feel free to ask more, and other people (more expert than me) can probably answer questions too

Ruby Khondaker (she/her) (Jul 09 2025 at 22:22):

thanks so so much!!

Chris Grossack (she/they) (Jul 09 2025 at 22:22):

ofc! ^_^

Ruby Khondaker (she/her) (Jul 09 2025 at 22:26):

Hm so are model categories only helpful for $(\infty, 1)$ categories?

Ruby Khondaker (she/her) (Jul 09 2025 at 22:30):

ooh actually maybe the tensor product might help here

Ruby Khondaker (she/her) (Jul 09 2025 at 22:38):

what i mean specifically is - i've seen papers talking about the model structure on complicial sets or comical sets

Ruby Khondaker (she/her) (Jul 09 2025 at 22:38):

but, does this present the full $(\infty, \infty)$ category of such objects? or only the truncated $(\infty, 1)$ category?

Ruby Khondaker (she/her) (Jul 09 2025 at 23:35):

Also if I’ve understood you correctly - model category structures are useful precisely when your objects aren’t all already bifibrant, right?

James Deikun (Jul 10 2025 at 04:48):

Model categories only model $(\infty,1)$-categories, not $(\infty,n)$-categories for higher $n$.

James Deikun (Jul 10 2025 at 04:49):

One less-obvious reason model categories are useful is that when you have a model category where not all objects are fibrant, or not all objects are cofibrant, and you only care about maps into/out of an object, you can use an object that's only fibrant/cofibrant but is much simpler than its cofibrant/fibrant replacement.

Mike Shulman (Jul 10 2025 at 04:51):

I wrote in the other thread:

Mike Shulman said:

Ruby Khondaker (she/her) said:

is there a way to get the $\omega$-category of such $\omega$-categories from this though

Not as easy a one. 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.

Mike Shulman (Jul 10 2025 at 04:53):

Ruby Khondaker (she/her) said:

Also if I’ve understood you correctly - model category structures are useful precisely when your objects aren’t all already bifibrant, right?

They're certainly more useful in that case, but I wouldn't say they're useless even when all objects are bifibrant, such as for the canonical model structure on Cat. For instance, in good cases model structures on a category $C$ can be lifted to functor categories $C^D$, or to the category of algebras for some monad on $C$, and in general these induced model structures will no longer have all objects bifibrant even if that was the case in $C$, so the original model structure on $C$ was useful as a way to get to these other ones.

Ruby Khondaker (she/her) (Jul 10 2025 at 06:29):

Oh ok, I see! So even if a “whitehead” theorem holds where your weak equivalences are the same as strong equivalences, it can still be good to single out a class of cofibrations and a class of fibrations

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fosco (Jul 13 2025 at 10:02):

Chris Grossack (she/they) said:

When working with oo-categories, there's many concepts that are only defined "up to a sequence of higher homotopies". This makes it almost impossible to do concrete computations (eg. limits/colimits, but already computing the space Hom(X,Y) can be difficult), since it's cumbersome to have to handle this sequence of higher homotopies by hand

The thread progressed too much for me to read everything, but a few days ago I wanted to say: this is an ex-post reading of the definition that cannot possibly have been what Quillen had in mind. So, what do you mean by this?