How do we see that Yoneda embedding preserves limis? Please see mathoverflow.net/questions/321474/…

@PraphullaKoushik Could you elaborate on your background? From your past questions I have the strong impression that you're lacking some kind of prerequisites for the math you're trying to learn, and figuring out where the gap is can help us compose a helpful answer.

If you just want a glib answer, the Yoneda embedding preserves limits simply because (homotopy) limits in function categories are computed pointwise (but I cannot imagine anyone understanding the above sentence and not knowing it already)

Another way to see that Yoneda preserves limits, which is the same as what Denis said but maybe in less fancy language, is that Yoneda replaces an object M with maps into M. But if M is a limit, then the universal property of a limit means that a map into M is the same as a compatible choice of maps into the diagram that M is a limit of. Then if you write down this statement in mathematical notation it's quite literally the statement that Yoneda commutes with limits

@DenisNardin I do not know anything about homotopy limit.. This term was mentioned by some one in another answer as something related to fibre product of morphism of Lie groupoids.. I know about Yoneda embedding...

Do you know how to prove that the Yoneda embedding preserves limits in classical categories?

it's by definition of limit, pretty much

@HarryGindi I know, but I suspect it would be beneficial if Praphulla tried to understand the proof in some detail :)

no denis i thought you had no idea ;)

@DenisNardin Yes Yes... I know that... In classical categories... Given a collection of objects C_i, limit is usually denoted by prod C_i and Yoneda embedding is just the hom functor and it preserves the product...

@PraphullaKoushik Well, you see, sentences like this make me suspect you have gaps in your background because you seem to be treating "products" and "limits" interchangeably, while they are different concepts. How well do you know classical category theory?

@DenisNardin Yes, It is necessary and beneficial for me to actually understand in detail..

If you think knowing a bit more category theory would help you, I suggest you read and do the exercises of the first five chapters in MacLane's Categories for the working mathematician. There might be other resources, but that is the one I learned from

In fact, even the first four chapters might suffice, as a start

@DenisNardin I learned some category theory from Hilton and Stammbach's book... It was an year and half ago and I did not use that recently so I am not so confident,,,

@DenisNardin I take your comment positively (I have to take if I want to become better).. I am not trying to defend my self here. My background in Category theory is little shaky... I will try to re read category theory... In what sense "product" and limit are different? I am afraid, the more I say, the more dumb I sound but can you please tell me what is the problem..

I don't mean to accuse you. I'm trying to help you figuring out how to solidify the understanding of the material you're trying to learn

That said, a product is a special kind of limit. It is in fact the limit of a discrete diagram. But not all limits are products: one example is the fiber product

@DenisNardin I do not take it as if you are accusing... :) :)

Recall that if C is a category, and I is a small category, an I-shaped diagram is just a functor F:I→C

@DenisNardin :O The case of pull back as in my question it is discrete right? Only two objects, arrows.. There limit and product are same right?

No, when I say discrete I mean that the category $I$ has only identity arrows

The category indexing a fiber product is $1→0←2$, which has two arrows that are not identities (the arrow 1→0 and the arrow 2→1)

In particular a fiber product is not a product (sorry... I didn't invent the terminology)

@DenisNardin (sorry... I didn't invent the terminology) :D

I will try to read about limits and products tonight.. I have done that before.. If I spend some more time, I will be able to get ir correctly.. you do not hav eto take pain of explaining :) I can ask after reading If I misunderstand anything

Denis, fwiw, it's not quite as obvious that the Yoneda embedding preserves holims because holims have about a dozen different definitions that are nontrivially related to one another

probably for 2-categories it's not so hard but general holims it's maybe easier if you know you can represent them by a weighted diagram etc etc

otherwise you end up deep into eg chapter 17 of hirschhorn reading about homotopy function complexes

@HarryGindi I know, but I wanted to know if they had at least the intuition why it was supposed to be true. If you don't understand the case of 1-categories there's little point in trying to figure out how homotopy limits are computed in functor categories...

@DenisNardin It is making sense why you are saying limit and product are not same... To define product you have a collection of objects A_i in a catgeory C and no arrows between two different indices is given.. This is like giving a functor Set---> C.... This matches with what you said that the Set has only identity arrows so there is no arrow from A_i to A_j for i\neq j ... Then there is notion of Product

Careful: this is not like giving a functor Set→C, but a functor I→C where I is a set :)

(Set is the usual name for the category of sets)

By Set i mean I is a set not the categoyr of sets..

@DenisNardin absolutely

Now,, coming to fibre prodict

you have three objects A_1, A_2 and A_3 now you have arrows A_1 to A_2, A_3 to A_2 ... These should be seen as coming from arrow 1--->2 and 3--->2 which is different from the case of product

both are limits of a functor... One where the index categroy has only trivial arrows and other has non trivial arrows..

Does this make sense? :O Or is it still not sensible?

let me know if it is not sensible, I would then open some text on Category theory... @DenisNardin I have written what every I can recollect...

@PraphullaKoushik Yes, this is correct

@DenisNardin Thanks :D I was afraid for some time that I do not know any category theory... It is not strong but not bad... :) I will anyways read category theory every now and then as you suggested...

Thanks for your time...

@PraphullaKoushik I was not really sure whether to add the tag to your recent question (since this is not really my area) - so I will just mention that there is also a tag called homotopy-limits.

@DenisNardin Please have a look at mathoverflow.net/a/321486/118688

(Admittedly, the tag is not very populated so far.)

@DenisNardin I shamelessly confess that if you have not said "I'm no expert on differentiable stacks, but I find hard to believe that someone can understand differentiable stacks well without a solid background on the basics of category theory" I would not have tried spending time on that... Thanks... :) :)

@MartinSleziak Thank you as always... Some user said it is homotopy limit but I am not really sure what exactly it mean... I can add that tag.. :) I have added that tag

@HarryGindi what book of hirschhorn are you talking about?

Model Categories and their localizations

@HarryGindi Thanks. What is that I need to know before understanding homotopy limit..

understanding homotopy limits is way harder than defining them

I tried google and saw some article by emily Riehl math.jhu.edu/~eriehl/hocolimits.pdf which need Kan extension and all that..

I dunno man

you don't need it in hirschhorn's setting

but it has a different meaning; I believe emily does it for simplicially enriched categories

Ok. Ok... If you get some time, can you look at mathoverflow.net/questions/321474/…

I don't have anything to say about it. I saw your question.

Ok Ok. Never mind. I was able to make some sense in classical categroy theory set up, thanks to Denis.. I could not figure out how it goes in the $2$-category set up..