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It might be a stupid question. Suppose There is a category of categories,denoted by CAT,where objects are categories, morpshims are functors between categories

Take multiplicative system S={category equivalences}. Then we take localization at S. Then we get localized category S^(-1)CAT

Another Category, denoted by |CAT|,where objects are categories, morphism are isomorphism classes of functors between categories.

I want to prove this two categories are equivalent. I want to prove this problem in two ways:

  1. Naive prove Because either |CAT| or S^(-1)CAT has same objects as CAT. So one can prove the morphism class of these two category has same equivalent relationship. For the S^(-1)Cat, suppose F and G are two functors in the same equivalent class, we have sF=sG,for s belongs to S. Then I can prove F and G belongs to the same equivalent class in |CAT|. But on the other hand, if F and G are isomorphic functor. I can not prove sF=sG,for some s belongs to S. What I can only prove is sF is isomorphic to sG.

  2. Using adjoint functors Because I want to prove the projection functor CAT--->|CAT| is a localization functor at S. So I want to construct an adjoint functor |CAT|--->CAT which is fully faithful. But what is this functor. It seems I can not make it well defined.

This is an exercise in Toen's lecture: Lectures on DG-categories

I attach the lecture notes on DG-categories here Toen notes

page 5,exercise 2

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4 Answers

up vote 7 down vote accepted

There are set theoretic issues that will hinder any proof of this statement. In particular I don't believe you can construct the adjunction in part (2) without applying the axiom of choice to CAT, and it's not clear the localization S^(-1)CAT makes sense either.

That said, let's ignore these issues and pretend that CAT is a small category of (small) categories.

I was originally confused by this. I believe that there are some problems in the formulation of this question. In particular if S consists of those functors which are equivalences of categories, then it does not form a multiplicative system. Specifically, the right Ore condition fails. I'll give an example later. What this means though, is that the property your are trying to check in part (1) actually fails. To see this pick a group G and view it as a category with one object. Then the functors from H to H are the homomorphisms, and the equivalence classes of this functors are the orbits of Hom(H,H) under the conjugation action. Choose H such that there are distinct automorphisms of H which are equivalent under conjugation.

Claim: If F,G: H --> H are two such automorphisms, then there is no equivalence s:H --> C such that sF = sG. I think this is easy to check, so I'll leave it at that.

So the second part of (1) is impossible to prove.

I first thought that we can answer this question be going to skeletal categories. This is not correct, but let's look at why it is not correct. Let SKEL be the full subcategory of CAT consisting of the Skeletal categories, i.e. those categories in which $a \cong b$ implies $a = b$, i.e. those categories which have a single object in each isomorphism class.

Every object in CAT is equivalent to one in SKEL. This is an easy exercise you should work out for yourself. By applying the axiom of choice to all of CAT we can construct an equivalence between CAT and SKEL, and in particular we have L: CAT ---> SKEL with an equivalence $C \cong L(C)$ for every category C.

Notice that a functor between skeletal categories is an equivalence if and only if it is an isomorphism on the nose. So we are part way there. This made me think that SKEL was the localizing subcategory we were after.

However two functors into a skeletal category can be naturally isomorphic without being equal. The group example above is actually an example of this. The functors F and G are naturally isomorphic, but not equal.

Let J denote the "Joyal interval", the category with two objects and an isomorphism between them. J can be used to say when two functors are naturally isomorphic. F,G: C --> D are nat. isomorphic if they extend to a functor $C \times J \to D$. There is a quotient of J where we identify the two objects. This is in fact the category Z (the group of integers viewed as a category). Let W be the set of morphisms consisting of the single morphism J --> Z. Notice that SKEL consists of exactly the "W-local" objects. But this is not quite what we want.

Okay, now for the rest. First, I promised an example where the right Ore condition fails. This is given by considering the two inclusions of pt into J. Both of these are equivalences, but this cannot be completed to an appropriate square. So the class of equivalences in not a multiplicative system.

Next, Toen doesn't assume that S is a multiplicative system. He (correctly) defines the localization in terms of a (weak) universal property. Namely, there exists a locization functor:

$ L: CAT \to S^{-1}CAT$

such that fro any other category D, $L^* : Fun( S^{-1}CAT, D) \to Fun(CAT, D)$ is fully-faithfull and the essential image consists of those functors which send elements of S to isomorphisms in D. In particular the quotient functor from CAT to |CAT| does this, so this gives us functor from S^(-1)CAT to |CAT| defined up to unique natural isomorphism.

On the other hand, |CAT| is also defined by a universal property. This gives us maps between |CAT| and S^(-1)CAT and by the usual sort of general nonsense this is an equivalence. The details aren't too hard, so I'll leave them to you.

Now the second part of your question asks about whether we can realize |CAT| as a full subcategory of CAT such that the quotient $ q:CAT \to |CAT|$ is left adjoint to the inclusion. The answer is no. Let's first look at whether q can have a right adjoint at all and what this would mean.

Suppose that R: |CAT| --> CAT is right adjoint to the quotient q. Be abuse, we will identify the objects of CAT and |CAT|. This means that for all categories X and Y we have:

$CAT(X, R(Y)) = |CAT|(X, Y) = |CAT|(X \times J, Y) = CAT( X \times J, R(Y))$

So $R(Y)$ must be cartesian local with respect to the map J --> pt. In particular we see, taking X=pt, that R(Y) has no non=identity automorphisms, i.e. R(Y) is rigid.

Now let's look at an example. Consider the group Z viewed as a category with one object. Then the endomorphisms of Z in |CAT| is the monoid Z (under multiplication). Now if R is fully-faithful then we need:

$ \mathbb{Z} = |CAT| (\mathbb{Z}, \mathbb{Z}) = CAT (R(\mathbb{Z}), R(\mathbb{Z}))$

but this last is equal to the automorphism of some set R(Z), hence we have a contradiction. So there is no such fully-faithful right adjoint.

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btw, I would denote your "|CAT|" by "hCAT", the homotopy category of CAT. –  Chris Schommer-Pries Dec 29 '09 at 12:28
"Two functors into an skeletal category are naturally isomorphic if and only if they are equal". Are you sure about this? Let $F$ and $G$ be one object groupoids, also known as groups. A functor between them is a morphism of groups. I think that two maps of groups give naturally isomorphic functors iff they are CONJUGATE, and give equal functors iff they are EQUAL. –  David Speyer Dec 29 '09 at 14:00
Hi David. After looking at it again, realized this was totally a false claim. I was editing while you made this comment. –  Chris Schommer-Pries Dec 29 '09 at 14:49
Out of curiosity, do equivalences obey the left Ore condition? (I am using the same handedness conventions as the nLab page you link to, which I believe are the same as yours.) –  David Speyer Dec 29 '09 at 16:37
I'm using the n-lab convention. No, I don't think they satisfy the left Ore condition either. I think the following is a counter example. We have an equivalence J --> pt, and a quotient map J --> Z (here the integers are viewed as a cat with one object). Since Z is skeletal an equivalence Z --> C is a faithful embedding, so WLOG we can assume C = Z. But the map J --> Z is surjective on morphisms, so it can't factor through the map J --> pt. Hence this diagram can't be completed to a correct square either. –  Chris Schommer-Pries Dec 29 '09 at 16:50
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Bertrand Toën is using a different definition of localization than what you describe in your question. Let $\mathcal{C}$ be a category and let $\Sigma$ be a class of morphisms in $\mathcal{C}$. A category $\mathcal{D}$ with a functor $L\colon \mathcal{C} \rightarrow \mathcal{D}$ is called a localization of $\mathcal{C}$ at $\Sigma$ if $L(f)$ is an isomorphism for every $f \in \Sigma$, and $L$ is universal with this property, that is, whenever $F \colon \mathcal{C} \rightarrow \mathcal{E}$ is a functor such that $F(f)$ is an isomorphism for all $f\in \Sigma$, then there exists a unique functor $G \colon \mathcal{C} \rightarrow \mathcal{E}$ with $F=G \circ L$. If a localization of $\mathcal{C}$ at $\Sigma$ exists, then it is unique up to isomorphism and ususally denoted by $\Sigma^{-1}\mathcal{C}$ or $\mathcal{C}[\Sigma^{-1}]$.

Note that we do not make any assumptions about the class $\Sigma$. As with all universal properties, such an object need not exist in the most general case: usually the resulting category might be "too big", that is, there would be a proper class of morphisms between certain objects in the localization. If the class $\Sigma$ admits a right calculus of fractions in the sense of (e.g.) Borceux, Handbook of Categorical Algebra, Volume I, Chapter 5, then the localization of $\mathcal{C}$ at $\Sigma$ does exist. However, there are lots of examples of categories with classes $\Sigma$ which do have a localization but don't admit a right calculus of fractions. In many of these cases, the functor $L \colon \mathcal{C} \rightarrow \mathcal{D}$ will then not have a right adjoint.

One of the main sources of such categories are Quillen model categories $\mathcal{M}$, with $\Sigma$ the class of weak equivalences. The localization at $\Sigma$ is then called the homotopy category of $\mathcal{M}$. In fact, the notion of Quillen categories was invented mainly to deal with the difficulties of formally inverting a class of arrows in a category. There is a model structure on the category of small categories (sometimes called the folk model structure) whose weak equivalences are the equivalences of categories, so the desired localization does exist. Moreover, there is a notion of left and right homotopy in a model category. For nice objects (objects which are both fibrant and cofibrant), these notions agree and define an equivalence relation on maps. In the model category of categories, all objects are fibrant and cofibrant. Two functors are homotopic if and only if they are naturally isomorphic.

The usual construction of the homotopy category of a model category (see e.g. Mark Hovey's book "Model Categories") shows that the localization of $\mathbf{Cat}$ at the equivalences is given by the category whose objects are again categories and whose morphisms are homotopy classes of morphisms. Thus we do indeed find that the localization of $\mathbf{Cat}$ at the class of equivalences is the category $\vert \mathbf{Cat} \vert$ of categories and isomorphism classes of functors, as Toën claimed.

Charles Rezk has a nice writeup of the folk model structure here.

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This is correct, but you don't need a whole model structure to see it. Every category C is equivalent to $C\times J$, where J is the free isomorphism, and a natural isomorphism between functors $C\to D$ is the same as a functor $C\times J\to D$. So if we quotient by natural isomorphism, then clearly equivalences of categories become isomorphisms, while if we make equivalences into isomorphisms, then $C$ becomes isomorphic to $C\times J$ and thus naturally isomorphic functors get identified. So the universal properties are the same, hence the resulting categories agree. –  Mike Shulman May 18 '10 at 4:05
BTW, the nLab is pushing to eradicate the phrase "folk model structure" in favor of "canonical model structure." –  Mike Shulman May 18 '10 at 4:06
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I'm pretty sure this is false. If I recall correctly, the localization S-1CAT you refer to is equivalent to the category of rigid categories, those with no nonidentity isomorphisms at all. In particular, the "rigid categorification" of any connected groupoid is the terminal category. On the other hand, the isomorphism classes of objects in |CAT| are equivalence classes of categories, so in |CAT| there are many nonisomorphic connected groupoids.

I'll see if I can remember how to prove my claim about S-1CAT, or maybe you can prove it; I don't think it's very hard, and it uses the same kind of ideas Chris talked about in his answer.

Oh, Toën is using a different definition of "localization" than the one I am accustomed to (e.g., see Definition and Warning of Higher Topos Theory). Since Cat is a presentable category, I assumed you were taking the localization in presentable categories, which involves a category S-1Cat and an adjunction with left adjoint Cat → S-1Cat which has the analogous universal property with respect to adjunctions between presentable categories. In that case, S-1Cat turns out to be rigid categories and Chris explained why you might expect this to be true. (The homotopy category of Cat is presumably not a presentable category at all.)

With S the class of equivalences of categories these localizations do not agree because S is not closed under pushouts in Cat: in the square

J -> BZ
|    |
v    v
* -> *

the left-hand map is in S but the right-hand map is not and yet the square is, bizarrely, a pushout. (What makes things even trickier is the ∞-categorical analogue of "S is closed under pushouts" is true because of course we must take homotopy pushouts. In short thinking of Cat as a 1-category and trying to work with equivalences there can lead to strange results!)

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thank you very much! –  Shizhuo Zhang Dec 29 '09 at 18:08
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One other way around the fact $S$ is not a multiplicative system is to use the approach of Dorette Pronk (in JPAA, 1996), namely looking at bicategories of fractions. Then one can show that the 2-category of small categories, functors and natural transformations, admits a bicategory of fractions with respect to the class $W$ of fully faithful, essentially surjective functors.

This is rather boring (as arrow in $W$ have pseudoinverses in Cat) unless you don't have the axiom of choice, when not every arrow in $W$ has a pseudoinverse. This was the approach by Makkai in his paper 'Avoiding the axiom of choice in general category theory', even if he doesn't reference Pronk. Pronk doesn't consider this example as she is working with topological and algebraic groupoids.

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