For a topological group $G$, we can construct the classifying space $BG$ as the geometric realization of the nerve of $G$. I have seen a very similar assertion in the context of algebraic geometry: For a reductive group $G$, the quotient stack $BG:=[*/G]$ can be represented as the homotopy colimit
$$ BG = \mathrm{hocolim} (* \leftleftarrows G \cdots ),$$
and this classifies principal $G$-bundles on schemes.
In this answer, this construction is called the bar construction. Why does this construction give the classifying space in both the topological and algebraic contexts? Is there a way to view these two situations as essentially the same?
Best Answer
In case it was not clear yet to you, let me note that the construction $BG\simeq |N(\mathbf{B}G)|$ (where $\mathbf{B}G$ is the one-object category with unique hom-set $G$) also witnesses the homotopy type $BG$ as homotopy colimit of the diagram $\Delta^\mathrm{op}\to\mathsf{Top}$ given by $$ \ldots \rightrightarrows\rightrightarrows G\times G\rightrightarrows\to G\rightrightarrows * $$ This means that both the classifying space $BG$ and the classifying stack $BG$ can be written as the same homotopy colimit, just in a different category. In fact, for a general simplicial set $X$, the homotopy type of the geometric realization $|X|$ is the homotopy colimit of the diagram $X\colon \Delta^\mathrm{op}\to\mathsf{Set}\hookrightarrow\mathsf{Top}$.
Now, let's see why we can expect this homotopy colimit to actually classify $G$-torsors (the definition will follow). For homotopical simplicity, I work in $\infty$-categories, but if you're at moments not comfortable with this you can just think about model categories and homotopy colimits in them, or about (2,1)-categories and (2,1)-colimits in them. Below, I will put references to Lurie's Higher Topos Theory, which I abbreviate to HTT.
The main intuition is written in the section ''The main theorem'' below. Let me know if this suffices for you. Unfortunately, to get to a good covering of the theory, I feel like I need to introduce quite a lot of notions, so I'm sorry if this is a bit too lenghty for your liking. You may want to first read the section ''The main theorem'', given that you probably have an intuitive idea what things like group objects, group actions and torsors should be. I prove the main theorem below for two reasons: you may want to see that, and I actually don't know where this result is proven for general $\infty$-topoi in reasonable detail and have been curious about the details since first learning about it, so this was a good opportunity to finally figure out some details. I don't write it up in full detail for reasons of length.
Definitions, part 1.
In a general $\infty$-category $\mathscr{C}$ with finite limits, we can define a groupoid object as a certain functor $U\colon\Delta^\mathrm{op}\to\mathscr{C}$ as in (HTT, 6.1.2.7) using the description in (HTT, 6.1.2.6(4'')). In simple yet slightly imprecise language, $U$ is a groupoid object if we can write it in the form $$ \ldots\rightrightarrows\rightrightarrows U_1\times_{U_0} U_1\rightrightarrows\to U_1\rightrightarrows U_0. $$ You can think about $U_0$ as the objects of the groupoid object, and $U_1$ as its morphisms, but this interpretation will not really be important for us here. A group object of $\mathscr{C}$ is a groupoid object $U\colon\Delta^\mathrm{op}\to\mathscr{C}$ such that $U_0\simeq *$ (i.e. a group is a groupoid with a single object). We write $\mathrm{Grpd}(\mathscr{C})$ for the full subcategory of $\mathrm{Fun}(\Delta^\mathrm{op},\mathscr{C})$ on groupoid objects.
Given a group object $G\colon\Delta^\mathrm{op}\to\mathscr{C}$ and an object $Y\in\mathscr{C}$, an action of $G$ on $Y$ is the data of a groupoid object $U$ in $\mathscr{C}$ and a morphism $U\to G$ of groupoid objects such that $U_0\simeq Y$ and such that every morphism $\alpha\colon [n]\to[m]$ in $\Delta$ with $\alpha(0)=0$ gives us a pullback diagram $$ \require{AMScd} \begin{CD} U_m @>{\alpha^*}>> U_n\\ @VVV @VVV\\ G_m @>{\alpha^*}>> G_n \end{CD} $$ In simple and again slightly imprecise language, an action of $G$ on $Y$ is a groupoid object of the form $$ \ldots\rightrightarrows\rightrightarrows G\times G\times Y\rightrightarrows\to G\times Y \rightrightarrows Y $$ A $G$-equivariant morphism between two such action groupoids $U$ and $V$ (with possibly non-equivalent $U_0$ and $V_0$) is a morphism of groupoids commuting with the structure maps $U\to G$ and $V\to G$, so we write $\mathrm{Grpd}_G(\mathscr{C})$ for the full subcategory of $\mathrm{Grpd}(\mathscr{C})_{/G}$ for the full subcategory on obects $U\to G$ where $U$ is an action of $G$ on some object of $\mathscr{C}$.
The trivial action of $G$ on $Y$ is the groupoid object $Y^\mathrm{triv}$ given by $[n]\mapsto G_{n}\times Y$, where $Y$ just tags along and all the structure maps come from those on $G$. In particular, the group object $G$ itself also encodes the trivial $G$-action on the terminal object $*$, so $G\simeq *^\mathrm{triv}$.
Definitions, part 2.
Given a morphism $f\colon X\to Y$ in $\mathscr{C}$, its Čech nerve is the groupoid object $Č(f)$ of $\mathscr{C}$ informally given by $$ \ldots\rightrightarrows\rightrightarrows X\times_Y X\times_Y X\rightrightarrows\to X\times_Y X\rightrightarrows X $$ Write $\Delta_+$ for the augmented simplicial category (obtained by adjoining a terminal object $[-1]=\varnothing$ to the category $\Delta$). The Čech nerve admits a canonical extension to an augmented simplicial object $\Delta^\mathrm{op}_+\to\mathscr{C}$ by putting the map $f\colon X\to Y$ at the end, which we write as $Č(f)\to Y$.
Given a simplicial object $U\colon\Delta^\mathrm{op}\to\mathscr{C}$, its geometric realization is the colimit $|U|:\simeq\mathrm{colim}_{\Delta^\mathrm{op}}\,U$ in $\mathscr{C}$. We will from now on assume $\mathscr{C}$ has all necessary colimits, and since we will be moving to $\infty$-topoi anyway in the future, let us just assume that $\mathscr{C}$ is complete and cocomplete from now on. Notice that we will later define $BG:\simeq|G|\simeq|*^\mathrm{triv}|$. The canonical map $U_0\to |U|$ gives us also an augemented simplicial diagram in $\mathscr{C}$ extending $U$.
An effective epimorphism is a map $f\colon X\to Y$ in $\mathscr{C}$ for which the Čech nerve $Č(f)$ exists (which it does by assumption that $\mathscr{C}$ is complete), and such that $f$ is equivalent to the canonical map $X\to |Č(f)|$. You can think about an effective epimorphism as a map $X\to Y$ that witnesses $Y$ as the quotient of $X$ under some equivalence relation on $X$. We write $\mathrm{EffEpi}(\mathscr{C})$ for the full subcategory of the arrow category $\mathrm{Fun}([1],\mathscr{C})$ on the effective epimorphisms. (Here, 1 denotes the one-arrow category $0\to 1$.)
Definitions, part 3.
Finally, given a group object $G$ in $\mathscr{C}$ and an object $X\in\mathscr{C}$, a $G$-torsor over $X$ is the data of an object $Y\in\mathscr{C}$, a groupoid object $U$ encoding a $G$-action on $Y$, and a $G$-equivariant morphism $\pi\colon U\to X^\mathrm{triv}$, such that $\pi_0\colon U_0\simeq Y\to X^\mathrm{triv}_0\simeq X$ is an effective epimorphism, and the canonical map $U\to Č(\pi_0)$ is an equivalence. Note that the data of $Y$ is redundant in the sense that it is encoded as $U_0$ already, but it helps me to think about the ''underlying object'' of a torsor instead of just a groupoid object. We denote by $\mathrm{Tors}_G(X)$ the full subcategory of $\mathrm{Grpd}_G(\mathscr{C})_{/X^\mathrm{triv}}$ on the $G$-torsors over $X$. We define $\mathrm{Prin}_G(X):=\mathrm{Tors}_G(X)^\simeq$, where $(-)^\simeq$ is the core of a category (so the maximal sub-$\infty$-groupoid). So in $\mathrm{Prin}_G(X)$ we still have all the $G$-torsors over $X$, but only remember the morphisms between them that are equivalences.
What we need about $\infty$-topoi.
We will from now on assume $\mathscr{C}$ is an $\infty$-topos in the sense of Lurie. It is not important what exactly an $\infty$-topos is, as we will only need that this implies that $\mathscr{C}$ is complete and cocomplete, that pullbacks commute with colimits in the sense that $X\times_Y -\colon \mathscr{C}\to\mathscr{C}$ commutes with colimits, and that there is an equivalence between groupoid objects and effective epimorphisms as follows.
Result A. If $\mathscr{C}$ is an $\infty$-topos, then every groupoid object $U\in\mathrm{Grpd}(\mathscr{C})$ gives an effective epimorphism $U_0\to |U|$ and every effective epimorphism $f\colon X\to Y$ gives a groupoid object via the Čech nerve $Č(f)$. It is a combination of (HTT, 6.1.0.6) and the text after (HTT, Cor. 6.2.3.5) that these constructions gives us inverse equivalences of $\infty$-categories $$ [\,(-)_0\to |-|\,]\colon \mathrm{Grpd}(\mathscr{C})\rightleftarrows\mathrm{EffEpi}(\mathscr{C})\colon Č(-). $$ This is very specific to $\infty$-topoi, but is the only deep theory about $\infty$-topoi that we need.
The main theorem.
Theorem. Let $G$ be a group object in an $\infty$-topos $\mathscr{C}$, and put $BG:\simeq |G|\simeq |*^\mathrm{triv}|$. For any $X\in\mathscr{C}$, there is an equivalence of $\infty$-groupoids $\mathrm{Prin}_G(X)\simeq \mathscr{C}(X,BG)$. Moreover, this equivalence is natural in $X$.
Intuitively, the equivalence looks like this: given a $G$-torsor $\pi\colon U\to X^\mathrm{triv}$ over $X$, note that, by definition, $U$ comes with a map $U\to G$. The induced map $X\simeq |U|\to |G|\simeq BG$ is the map we are after. The equivalence $X\simeq |U|$ comes from the fact that $\pi_0\colon U_0\to X$ is an effective epimorphism and that $U\simeq Č(\pi_0)$. Conversely, a map $X\to BG$ allows us to pull back the augmented groupoid object $*^\mathrm{triv}\to BG$ to an augmented groupoid object $U\to X$ over $X$, and since effective epimorphisms are stable under pullback, it is not hard to show that this will be a $G$-torsor over $X$.
Here lies the intuition why $BG$ classifies $G$-torsors, and why it had to be defined like this: any map into $BG$ gives via pullback a $G$-torsor, as $BG$ comes equipped with a canonical $G$-torsor $*^\mathrm{triv}$, but more importantly, $*^\mathrm{triv}$ is almost the terminal torsor in the sense that any other $G$-torsor over some object must come with a ''unique'' map towards $*^\mathrm{triv}$ by definition of a $G$-action. (This map has to be the structure map $U\to G$, but the space of $G$-torsor maps $U\to *^\mathrm{triv}$ fails to be contractible, so $*^\mathrm{triv}$ is not actually the terminal torsor. For our purposes, it's close enough to have just a canonical map, and it's even a nice feature that it comes with nontrivial self-homotopies.)
Now, if you think you have a classifying object for torsors, it sort of has to be given to you by the object admitting the ''terminal'' torsor. Given a $G$-torsor $U$ over $X$, we saw that $X\simeq |U|$, so our best guess is that the classifying object has to be given by $|*^\mathrm{triv}|$. For the actual proof that this works, you need that little bit of nontrivial mathematics about effective epimorphisms in $\infty$-topoi.
Turning both these constructions above into actual functors $L\colon\mathrm{Prin}_G(X)\rightleftarrows \mathscr{C}(X,BG)\colon R$ and showing they are natural in $X$ is not that difficult if you know enough of HTT, safe for one detail, so this is not really the interesting part. Instead, I'd like to prove that they are equivalences of $\infty$-groupoids. For this, we will actually need this one detail mentioned just before, which is the following: given two $G$-torsors $U$ and $V$ over $X$, we must know that an equivalence $g\colon U\to V$ of $G$-torsors gives on colimits the identity map $X\simeq|U|\to |V|\simeq X$, and not just any equivalence. The reason we need this is that we want $g$ to induce a homotopy between two maps $X\to BG$, and we can easily get a such a homotopy if we precompose one of these maps with $|g|$, but we will say more about this in a clearer way later.
The proof that $|g|\simeq\mathrm{id}_X$
We write $p\colon *\to BG$ for the canonical map coming from the definition $BG\simeq |*^\mathrm{triv}|$.
Lemma 1. Given a morphism $g\colon U\to V$ of $G$-torsors over $X$, the induced map $|g|\colon X\simeq |U|\to |V|\simeq X$ is equivalent to $\mathrm{id}_X$.
Proof of Lemma. The morphisms $\pi\colon U\to X^\mathrm{triv}$ and $U\to G$ that $U$ comes equipped with are compatible in the sense that $\pi$ is $G$-equivariant. Since $X^\mathrm{triv}\simeq X\times G$ as groupoid objects in a $G$-equivariant manner, the map $U\to X^\mathrm{triv}$ is completely determined by the map $\pi_0\colon U_0\to X$ and the structure map $U\to G$. Now, $|X^\mathrm{triv}|\simeq |X\times G|\simeq X\times |G|\simeq X\times BG$ (here we use that $X\times -$ commutes with colimits in $\mathscr{C}$ since the latter is an $\infty$-topos), so the map $\pi$ gives us a commutative square $$\require{AMScd} \begin{CD} U_0 @>{\pi_0}>> X\simeq X\times *\\ @V{\pi_0}VV @VV{\mathrm{id}_X\times p}V\\ X @>{\mathrm{id}_X\times L(U)}>> X\times BG \end{CD} $$ Since $g\colon U\to V$ has to commute with the structure maps to $X^\mathrm{triv}$, on colimits we find that the map $|g|\colon X\to X$ has to commute with both maps $\mathrm{id}_X\times L(U), \mathrm{id}_X\times L(V)\colon X\to X\times BG$. This forces $|g|\simeq\mathrm{id}_X$, and finishes the proof of the lemma.
So, this is useful because $g$ has to commute with the structure maps $U\to G$ and $V\to G$, and therefore we get by definition a homotopy $L(U)\simeq L(V)\circ |g|\simeq L(V)$. This is exactly a morphism in $\mathscr{C}(X,BG)$, which is part of the functoriality of $L$.
The proof that $L$ is essentially surjective.
We will see that $L$ is an equivalence, and we will start by showing that $LR(f)\simeq f$ for any $f\colon X\to BG$. It will be plausible from construction that these equivalences are natural in $f$ (I'm too lazy to write that out formally), so then $R$ will at the end via formal reasons also be an equivalence and an inverse to $L$.
Let $f\colon X\to BG$ be given. Then $R(f)$ gives us a $G$-torsor $X\times_{BG} *^\mathrm{triv}$ over $X$. Its structure morphism to $G$ is given by projection on the second component (as $*^\mathrm{triv}\simeq G$). Therefore $LR(f)$ is given by $|X\times_{BG} *^\mathrm{triv}|\to |G|\simeq BG$. Since $X\times_{BG} -$ commutes with colimits (as $\mathscr{C}$ is an $\infty$-topos, see above), we have an equivalence $|X\times_{BG} *^\mathrm{triv}|\simeq X\times_{BG} |*^\mathrm{triv}|\simeq X\times_{BG} BG\simeq X$. The last step shows that the map $X\to BG$ corresponding to the projection $X\times_{BG} BG\to BG$ is the structure map $X\to BG$ that we used to define the pullback. But this is just the map $f$ we started with. Therefore $LR(f)\simeq f$. Therefore, $L$ is essentially surjective.
Another lemma.
Now, we must see that $L$ is fully faithful. For this, we need the following lemma first.
Lemma 2. Suppose given a $G$-torsor $U$ over $X$, corresponding to the morphism $f\colon X\to BG$ under $L$. Then the diagram $$\require{AMScd} \begin{CD} U_0 @>>> *\\ @V{\pi}VV @VVV\\ X @>{f}>> BG \end{CD} $$ is a pullback diagram.
Proof of Lemma. We have to show that the canonical map $U_0\to X\times_{BG} *$ is an equivalence. It is (HTT, 6.2.3.16) that a morphism $h\colon A\to B$ in an $\infty$-topos is an equivalence when some pullback $A\times _B B'\to B'$ along an effective epimorphism $B'\to B$ is an equivalence. In other words, pulling back along an effective epimorphism is a conservative functor. Since $\pi_0\colon U_0\to X$ is an effective epimorphism, it therefore suffices to show that the canonical map $U_0\times_X U_0 \to U_0\times_X (X\times_{BG} *)\simeq U_0\times_{BG} *$ is an equivalence. In the latter pullback, the structure map $U_0\to BG$ is the composition $U_0\to X\xrightarrow{f} BG$. This composite equals the composite $U_0\to *\to BG$ because of the commutative square in the statement of the lemma. But the pullback of $*\to BG$ along the morphism $U_0\to *\to BG$ is the object $U_0\times(*\times_{BG} *)\simeq U_0\times G_1$. Here we used that $G_1\simeq *\times_{BG} *$, which generalizes the classical result that a group $G$ is the homotopy fiber of the universal cover $EG\to BG$ in topology (namely, given a group object $G$, the object $G_1$ is its underlying object, and the rest of the data is the data of homotopy coherent group structure). So why is $G_1\simeq *\times_{BG} *$? It is because $*\to BG$ is an effective epimorphism as geometric realization of $G$, so by Result A $G$ is the Čech nerve of $*\to BG$. But then, by definition, $G_1\simeq *\times_{BG} *$.
Okay, so we have found that it suffices to show that $U_0\times_X U_0\to U_0\times G_1$ is an equivalence. Going through the construction, it is informally given by $(u,v)\mapsto (u,g)$ where $g\in G_1$ is the unique element such that $gu=v$. From classical torsor theory, you might have expected that we would have required that the map $(\mathrm{id},\mathrm{act}) \colon G_1\times U_0\to U_0\times_X U_0$ is an equivalence, as this is how torsors are classically defined. And indeed, the requirement that $U\simeq Č(\pi_0\colon U_0\to X)$ gives us in degree 1 that this map $G_1\times U_0\to U_0\times_X U_0$ is an equivalence. Our map $U_0\times_X U_0\to U_0\times G_1$ in the other direction can be shown to be a one-sided inverse of this equivalence, and hence is an equivalence as well. This concludes the proof of the lemma.
The proof that $L$ is fully faithful.
Suppose given two $G$-torsors $U$ and $V$ over $X$. By definition, the space of morphisms between them is the space of morphisms $g\colon U\to V$ of groupoid objects in $\mathscr{C}$ that make the diagram
commute. By Result A and Lemma 1, this is equivalent to the space of maps $g_0\colon U_0\to V_0$ fitting in the diagram
where all dashed vertical morphisms are effective epimorphisms (they are dashed to improve readability). However, we know from Lemma 2 that the outer vertical squares involving $U_0$ (resp. $V_0$), $X$, $BG$ and $*$ are pullback squares, and $g_0\colon U_0\to V_0$ is a morphism that has to respect these pullback squares. As such, by the universal property of pullbacks, the space of morphisms $g_0$ fitting in the big diagram above is equivalent to the subspace of the space of morphisms between the two cospans as indicated below,
where we impose the restriction that on objects, we want identities, and the homotopies witnessing the commutativity of the top square also have to be identities. So the only data left to pick freely is the homotopy between $L(U)$ and $L(V)$ in the bottom square. This space is therefore equivalent to the space of homotopies between $L(U)$ and $L(V)$, and one may check that the chain of equivalences $\mathrm{Prin}_G(X)(U,V)\simeq \mathscr{C}(X,BG)(L(U),L(V))$ is given by $L$. Therefore, $L$ is fully faithful, proving that it is an equivalence.
Recovering the classical results.
In case $\mathscr{C}=\mathrm{An}$ is the $\infty$-category of anima/spaces/$\infty$-groupoids, the object $BG$ is (the underlying homotopy type of) the space that you usually encounter in topology, and indeed also given by $BG\simeq |N(\mathbf{B}G)|$. Hitting the equivalence $\mathrm{Prin}_G(X)\simeq\mathrm{An}(X,BG)$ by $\pi_0$ (i.e. going to homotopy categories), we find the classical result that homotopy classes of maps $[X,BG]$ are in bijection with isomorphism classes of $G$-torsors on $X$, although some care needs to be taken when you want to see $X$ as a topological space, rather than just a homotopy type.
It is generally a problem that $1$-topoi and $n$-topoi are not $\infty$-topoi. But in case $\mathscr{C}$ is something like stacks on the big étale site (and hence a $(2,1)$-category), the $\infty$-categorical definition of $BG$ also gives you the classifying stack as it is classically defined in stack theory as $\infty$-colimits in a $(2,1)$-category are just $(2,1)$-colimits. Moreover, the main theorem above and its proof also work for general $n$-topoi, using the analogous statements that we have for those, and since stacks form a $2$-topos, we get the classical classification result. You may enjoy figuring out what the classification result gives you in case $\mathscr{C}$ is the 1-topos $\mathsf{Set}$ of sets, or a 1-topos $\mathrm{Fun}(\mathcal{I}^\mathrm{op},\mathsf{Set})$ of presheaves. You should recover standard results about torsors of sets and torsors of presheaves, that you may not have expected to be related to $BG$ (although $BG$ is a bit of a silly object for $1$-topoi, the main theorem still holds for it, which is nice).