# The nerve-realization of $[n]\mapsto\Pi_1(\Delta^n)$

Consider the diagram

Where the functor $G$ sends a topological space to the category having as objects its points, and arrows homotopy classes of paths, $\varrho$ "realizes" geometrically an element of the simplex category $\Delta$ and $y$ is the Yoneda embedding.

I am interested in characterizing the nerve-realization adjoint pair induced by the composition $G\circ\varrho$, i.e. the adjoint pair $${\rm Lan}_{y}(G\varrho)\colon X\mapsto \int^{n\in\Delta}\widehat{\Delta}\big(\Delta[n],X\big)\cdot G\varrho([n])\quad\dashv\quad N_{g\varrho}\colon \mathbf C\mapsto {\bf Cat}(G\varrho(-),\bf C)$$ where I can simplify again ${\rm Lan}_y(G\varrho)$ by noticing that it is isomorphic to $\int^n X_n\cdot G(\Delta^n)$, where $\Delta^n$ is the standard topological simplex; it is extremely easy to notice that if $G$ has a right adjoint, say $\sigma$, then $N_{G\varrho}\cong \sigma^*N_\varrho$, where $N_\varrho$ is the classical "singular complex" functor (the nerve associated to $\varrho$).

Google does not give any clue about this problem, except for a reference to the adjunction between the "fundamental groupoid" (which is a different functor from my $G\varrho$) and the (restriction of the) classical nerve to the subcategory of groupoids, denoted $N_{\bf Gpd}$ in the following commutative diagram of adjoint pairs:

(if $(-)^\times\colon\bf Cat\to Gpd$ is the "maximal subgroupoid" functor, then the left adjoint to $N_{\bf Gpd}$ is obtained as the composition $X\mapsto (RX)^\times$, simply composing adjoints).

Has the functor $G$ an adjoint, in such a way that it respects realization (which is a colimit)? If it has not, can you still give me an explicit description for the pair nerve-realization associated to $G\varrho$? Can you tell how does this adjunction "descend to homotopy categories", if I put on both categories ($\bf Cat$ and $\widehat\Delta$) suitable model structures [better to say, if I put it on $\Delta$: if I remember well $\bf Cat$ is quite forced to have the structure (categorical equivalences, Grothendieck fibrations)]?

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$\mathbf{Cat}$ supports two distinct non-trivial model structures, neither of which have Grothendieck fibrations as fibrations, so I very much doubt it is "forced" to have any particular one! –  Zhen Lin Sep 2 '13 at 16:49
Hence I didn't remember well. :) –  tetrapharmakon Sep 2 '13 at 16:52
$G$ does not preserve colimits so it cannot have a right adjoint. However, it does preserve all colimits involved in gluing together simplices (this is essentially van Kampen). This means that your "realization" functor is just the composition of $G$ with the usual realization functor. –  Eric Wofsey Sep 2 '13 at 17:03
"$G$ does not preserve colimits so it cannot have a right adjoint. However, it does preserve all colimits involved in gluing together simplices (this is essentially van Kampen)" Thank you. Can you make this more precise (finding a colimit which is not preserved is maybe easy, but how can one characterize a "VK"-like colimit?) –  tetrapharmakon Sep 2 '13 at 17:22