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The correct construction for a topological category is as follows:

If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.

Now it suffices to give the answer for simplicial categories.

However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.

The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.

Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$

Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:

$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.

for any simplicial category $C$.

Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).

As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...

As for how they're related, Segal explains it himself on page 2 of this paper

Edit: Upon closer reading, I noticed that Segal's definition above is very close to the definition of a Segal category or Segal space. To show that the notions agree, we may be required to use the Quillen equivalence proven (by Joyal, I think?) that quasicategories are Quillen equivalent to segal categories and complete segal spaces.

The correct construction for a topological category is as follows:

If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.

Now it suffices to give the answer for simplicial categories.

However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.

The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.

Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$

Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:

$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.

for any simplicial category $C$.

Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).

As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...

Edit: Alright, so the reason why they agree is covered in §4.2.4 of HTT, I'm pretty sure.

The correct construction for a topological category is as follows:

If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.

Now it suffices to give the answer for simplicial categories.

However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.

The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.

Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$

Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:

$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.

for any simplicial category $C$.

Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).

As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...

As for how they're related, Segal explains it himself on page 2 of this paper

The correct construction for a topological category is as follows:

If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.

Now it suffices to give the answer for simplicial categories.

However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.

The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.

Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$

Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:

$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.

for any simplicial category $C$.

Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).

As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...

As for how they're related, Segal explains it himself on page 2 of this paper

Edit: Upon closer reading, I noticed that Segal's definition above is very close to the definition of a Segal category or Segal space. To show that the notions agree, we may be required to use the Quillen equivalence proven (by Joyal, I think?) that quasicategories are Quillen equivalent to segal categories and complete segal spaces.

The correct construction for a topological category is as follows:

If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.

Now it suffices to give the answer for simplicial categories.

However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.

The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.

Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$

Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:

$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.

for any simplicial category $C$.

Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).

As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...

The correct construction for a topological category is as follows:

If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.

Now it suffices to give the answer for simplicial categories.

However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.

The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.

Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$

Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:

$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.

for any simplicial category $C$.

Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).

As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...

As for how they're related, Segal explains it himself on page 2 of this paper