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What is a formal way to prove the fundamental groupoid functor (from both simplicial sets and spaces) preserves homotopy colimits?

I've tried looking online for such proofs but couldn't find any, and the answers I've received so far used facts whose proofs I also couldn't locate, e.g groupoids are Quillen equivalent to 1-types, truncations preserves homotopy colimits. I understand the proof probably relies on the compatibility of Quillen adjunctions with homotopy (co)limits.

If you answer, please be explicit and include details - I'm sure this is easy for experts, but for novices, it's very hard to fill in the blanks.

Thanks!

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    $\begingroup$ The model-independent answer is that taking the fundamental groupoid is left adjoint to the inclusion of 1-types into homotopy types. If you're looking for a model-dependent statement you should clarify e.g. what model of homotopy colimits you're using. $\endgroup$ Aug 25, 2016 at 18:42
  • $\begingroup$ @QiaochuYuan isn't your first sentence saying that $\Pi_1$ is "1-truncation"? But what exactly are the categories and results involved? $\endgroup$
    – Arrow
    Aug 26, 2016 at 10:29

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Let's take Kan complexes as our spaces. Then the fundamental groupoid functor $\Pi$ sends homotopic Kan complex morphisms to naturally isomorphic functors of groupoids. In particular, it sends homotopy equivalences of Kan complexes to equivalences of groupoids. You can prove this simply by seeing a homotopy between $f,g:K_1\to K_2$ as a map $K_1\times I\to K_2$ restricting to $f,g$. That analogous description reduces natural isomorphisms in groupoids to functors, so the claim follows as soon as one shows $\Pi$ preserves finite products, which is straightforward. Similarly, the nerve functor from groupoids to Kan complexes sends natural isomorphisms to homotopies, since $N$ as a right adjoint certainly preserves products.

So we have an adjunction $\Pi:\mathrm{Kan}\to \mathrm{Gpd}:N$ in which both adjoints preserve the appropriate notion of weak equivalence. This automatically gives rise to an adjunction between the homotopy categories. If this isn't clear, it's easy to see from the 2-universal property of localization, which people unfortunately often forget to state. The localization $\mathrm{Ho(Kan)}$ of Kan complexes at homotopy equivalences has the universal property that the category of functors $\mathrm{Ho(Kan)}\to D$ is isomorphic to the category of functors $\mathrm{Kan}\to D$ which send homotopy equivalences to isomorphisms. This gives, in particular, a functor $N\Pi:\mathrm{Ho(Kan)}\to \mathrm{Ho(Kan)}$ with a unit $\eta:1\to N\Pi$, and similarly $\Pi N$ and $\varepsilon$ on the $\mathrm{Ho(Gpd)}$ side, satisfying the triangle identities.

More generally, for every small category $J$ we have levelwiss fundamental groupoid and nerve functors $\Pi^J:\mathrm{Kan}^J\to \mathrm{Gpd}^J:N^J$. These are still adjoint (general fact, e.g. from 2-functoriality of exponentiation by $J$) and still preserve weak equivalences in the functor categories, which are levelwise homotopy equivalences of Kan complexes or equivalences of groupoids, respectively. Thus by the same argument from above, for every $J$ we have an adjunction $\Pi^J:\mathrm{Ho(Kan}^J)\to \mathrm{Ho(Gpd}^J):N^J$. Note this adjunction is 2-natural in $J$, again by 2-functoriality of the internal hom of categories.

Now the homotopy colimit functor is characterized as the left adjoint $\mathrm{hocolim:Ho(Kan}^J)\to \mathrm{Ho(Kan}):\Delta$, where the right adjoint $\Delta$ is the constant diagram functor: that is, homotopy colimits are derived colimits, Given a $J$-shaped diagram of Kan complexes $X$, we want to show $\Pi(\mathrm{hocolim}X)\cong \mathrm{hocolim}(\Pi X)$, where the right-hand $\Pi$ is levelwise. This is a case of the following general situation.

Given any 2-natural transformation $\alpha: F\to G$, we get a collection of squares indicating that $\alpha_aF(u)=G(u)\alpha_b$ for every $u:b\to a$ in the common domain of $F,G$. What happens to this square when we take left adjoints $F(u)_!,G(u)_!$ of the images of $u$? (This is exactly what we hope to do in the example at hand, when $u:*\to J$ is the unique functor in $\mathrm{Cat}^{\mathrm{op}}$, which induces the diagonal functor upon exponentiating by $\mathrm{Gpd}$ or $\mathrm{Kan}$.)

There is generally no commutative square of adjoints. What we do have is a 2-morphism $\xi:G(u)_!\alpha_a\to \alpha_bF(u)_!$ called the mate, given by $G(u)_!\alpha_a\to G(u)_!\alpha_aF(u)F(u)_!=G(u)!G(u)\alpha_bF(u)_!\to \alpha_bF(u)_!$, using the unit of $F(u)$ and the count of $G(u)$. Asking for $\xi$ to be an isomorphism is asking, in our example, exactly for $\Pi$ to preserve $J$-shaped colimits: $\xi$ is the canonical map $\mathrm{hocolim}(\Pi X)\to \Pi(\mathrm{hocolim} X)$.

So, why is our $\xi$ an iso? This has, again, a general 2-categorical answer. Suppose the $\alpha$s have right adjoints $\beta$. Then the conjugate of $\xi$ is a natural transformation $\zeta$ between the right adjoints of the domain and codomain of $\xi$. That is, $\zeta:\beta_bF(u)\to \beta_aG(u)$. By the definition of conjugate a (see Maclane, around page 98), $\xi$ is an isomorphism if and only if $\zeta$ is. You can check that $\zeta$ is the other mate of our square $\alpha_aF(u)=G(u)\alpha_b$, or read the general result about compatibility of mates with pasting in Kelly and Street's Elements of 2-categories.

$\zeta$ is then an iso if $\beta_bF(u)=\beta_aG(u)$, and if $\beta$ is right adjoint to $\alpha$ in the 2-category of arrows and commutative squares. This is probably unclear: this comes from another fundamental paper in 2-category theory, Kelly's Doctrinal Adjunction. The relevant theorem, proved in the first three pages of the paper, is that an adjunction in a 2-category lifts to a 2-category of algebras for a 2-monad, e.g. a levelwise adjunction lifts to a 2-category of 2-functors, if and only if the left adjoint admits a structure 2-isomorphism whose mate gives the structure of a pseudo-morphism to the right adjoint (For us, the left adjoint has a strict structure, since the original square commutes.) In our example $\beta$ is $N$, and the adjunction is, as mentioned above, sufficiently natural. Thus $\zeta$ is iso, so is $\xi$, and $\Pi$ preserves homotopy colimits.

Note we've really proved this is true for any weak equivalence-preserving adjunction of relative categories which admit homotopy colimits. This is also, in some sense, a model independent proof: any situation in which you can talk about homotopy colimits will satisfy the conditions needed to make this argument.

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