# 3.2. 群胚纤维范畴

We note in category theory, a set means a category where the only maps are identity (this is definition).

Note for such , we have a well-defined pullback map. Indeed, we get diagramThen we get lying over , i.e. is morphism in . But is a set, so it must be identity by definition, i.e. .

So, given , we get compatible with composition, i.e. naturally yields a presheaf where .

The next task is to show categories fibered in sets are the same as presheaves.

We note again that in higher category theory, sets are, by definition, categories with only identity morphism.

This is a fibered category because: we just let be , and we need to show pullback exists. Suppose we havewe claim this is a pullback, i.e. all maps are Cartesian.

To check this, we have the following diagramIn this diagram, we get and suppose we are given ,We are trying to show there is unique map , i.e. we want

Why this map exists? By definition, means as we are working with presheaf. Hence we indeed get the desired arrow .

We should also check is a set, so that is fibered in sets.

The objects of are with . The morphisms are lying over identity , i.e.but and hence . Thus morphisms in are identity, i.e. is a set.

We defined the maps already, so one should check these are quasi-inverse maps

Throughout the course, we will identify with .

We note in the proof, we only need being fibered in groupoids. Also, by the same argument, one can show that if is category fibered in groupoids, then all arrows are Cartesian.

Next we consider a nice example of categories fibered in groupoids.

So, here is the intuition: Here:

 1 : source 2 : target 3 : identity 4 : inverse 5 : multiplication/composition

Here is how you supposed to think of this set of data.

is supposed to be like objects in a category, is supposed to be arrows in the category. Then what’s going on with and inis that, takes an arrow and sends it to the source, while takes an a rrow and sends it to the target.

takes objects to the identity arrow.

takes an arrow to its inverse (this supposed to exists because its a groupoid).

is like an arrow such that the source of is equal the target of . In other word, under is supposed to be “”.

The above is the intuition, and let’s give the actual axioms about groupoids in .

Axioms:

 1 2 and . 3 4 5 (Associativity): the following two maps (we dropped the in all the fibered products here)are equal. 6 (Identity):This says and . 7 (Inverse): we get diagramsThis says that and .

Last time we end up listing the axioms of groupoids in . The definition seems complicated, but the idea is not bad. Basically, should be objects, be morphisms, takes morpihsms to its source, and to the target, sends objects to identity map, to inverse arrow, and is composition of arrows.

We remark that, groupoids are sort of generalization of groups, where groups in category consists of only one object, and groupoids consists of more than one objects.

Today we are going to show we can get from groupoids in to categories fibered in groupoids over .

Given , let be a category defined as follows: objects are , and morphisms for is an element such that and (here , and hence is an arrow , i.e. it make sense to ask and so on). This is a category, where composition of arrows are given by apply , i.e. say we have and , where is given by .

Next, we define a fibered category over as follows: objects are , where and (recall objects of this category is just ). To get the morphisms, note given , we get the arrow (which is a functor)is well-defined (i.e. induces and and hence ). Then, morphisms will be given by pairs and an isomorphism in .

Then the projection is going to be .

It remains to check is a category fibered in groupoids.

The fiber is the category defined by: objects are, by definition, just (as it is the same as the objects of ). The morphism for is given by and . However, since must live over the identity, and hence . In other word, morphisms are just , i.e. is a groupoid, as desired.

Aside, if is category fibered in groupoids, for we can define , which is a category fibered in groupoids where behaves like objects of over . This notion is hardly been used, so if we need it in the future we will define it, but for now its just aside.

The next notion is rather important.

Let in ,because is a groupoid (hence all arrows are isomorphisms). This depends on choices of pullbacks but we just fix one for all .

Why is a presheaf?

Say we have have our arrows , and . Then we getwhere and are two choices of pullback. However, note all pullbacks are isomorphic, we see that we get In particular, this means that is canonical isomorphism and hence we get (canonical) arrowwhich concludes is a presheaf (as it is compatible with composition of arrows).

Next, we define fibered products of groupoids. So, unlike normal fibered products in -category, now we are working with -categories, hence we also need to consider arrows between arrows.

We will start with a diagram of groupoidsWe are going to define so that we get the following diagram:where is arrow between arrows.

Next, we will define some arbitrary category , then we talk about universal properties that will convince us this is what fibered products should be for groupoids.

The objects are where and .

The morphisms for to will be a pair so that we get diagram

Next, we need to define and . They are given by and , and .

To add a few words on , we note , hence by definition this means we want that, inside , for any , we get the following commutative squareBut if you expand the definition of , this becomes exactly the square in definition of morphisms in . Hence is indeed natural trans as desired.

The universal property will be that, for all diagramwhere is a groupoid with , and isomorphism (which is natural transformation) , we get unique with diagramwhere the two arrows are natural trans between arrows (, respectively) and the composing arrows of and ( and , respectively), so that the diagramcommutes.

Well, why is this object exists? To answer this, we want to construct the unique . This will be the most “obvious” thing to do, which is . We left the details of how acts on morphisms, as it should be natural.

Now we want to make sure we get the natural transformations . That is, we want a natural trans . This is the same as, for arbitrary we want to get . Well, there is only one natural thing to do, which is take to be identity between and . We do the same for .

Next we need to check commutativity of the following diagramwhere is given by definition:Now for any , we getbut then in particular by definition. Hence it is indeed commutative.

This concludes the definition of fibered products of groupoids, and we are heading to define fibered products of categories fibered in groupoids.

For this, let be categories fibered in groupoids. Then we want to have a diagramso that for all and we get unique arrow with additional arrows between the arrows.

We want to have the property thatHowever, on the RHS, they are just fibered products of groupoids, and by -Yoneda lemma, this determines if the RHS exists.