If we have a category $\mathcal{C}$, then we can see it as an $\infty$-category. Furthermore, we can truncate and $\infty$-category $\mathcal{X}$ to get a category $\mathcal{X}_{\leq 1}$. My question is if these functors are adjoint, i.e. if we have $$\text{Hom}_{\mathfrak{Cat}}(\mathcal{X}_{\leq 1}, \mathcal{Y})\cong \text{Hom}_{\infty\text{-}\mathfrak{Cat}}(\mathcal{X},\mathcal{Y})$$ where $\mathcal{Y}$ is a category (which on the right is seen as an $\infty$-category).
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$\begingroup$ I think you need to edit a bit your question. What is Y? $\endgroup$– Mirco A. MannucciCommented Sep 19, 2020 at 20:28
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There is a bit of notation to be careful about here:
$\mathcal{X}_{\leqslant 1}$ is often used to denote the full subcategory of $\mathcal{X}$ of set-truncated object. For example if $\mathcal{X}$ is an $\infty$-topos, then $\mathcal{X}_{\leqslant 1}$ is its $1$-topos reflection.
with this definition, $\mathcal{X}_{\leqslant 1}$ is a $1$-category, but it is not the one that will have the property you want (it will be a right adjoint instead of a left adjoint, and only when restricted to finite limit preserving functor).
The $1$-category you want to consider is the homotopy category $h \mathcal{X}$ of $\mathcal{X}$, sometimes also denoted $\tau \mathcal{X}$, which is the category with the same objects as $\mathcal{X}$, and with the morphism sets
$$ h\mathcal{X}(a,b) \simeq \pi_0 ( \mathcal{X}(a,b) ) $$
Which does satisfies the property you ask.
A rigorous proof of this of course depends on what model of $\infty$-category you use, but if you use quasi-categories, this follows from points 1.2 and 1.8 in Joyal notes on quasi-categories.
answered Sep 19, 2020 at 20:53
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$\begingroup$ Just to clarify: the functor $(-)_{\leq 1}$ which restricts the fundamental $\infty$-groupoid to the fundamental groupoid of a space has no right adjoint? I'm asking because I would like to use this to give a simple proof of Seifert-van Kampen: For an open cover $U,V$ of $X$, we have a pushout $U\cap V\rightarrow U\times V\rightarrow X$. If we use the homotopy hypothesis, we get a pushout $\Pi(U\cap V)\rightarrow \Pi(U)\times \Pi(V)\rightarrow \Pi(X)$. If the truncation functor $\tau_{\leq 1}(-)$ were adjoint, then the pushout would be preserved, yielding a nice proof of S-vK. $\endgroup$ Commented Sep 19, 2020 at 21:31
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3$\begingroup$ The $\pi_1$ functor taking an $\infty$-groupoid to its fundamental groupoid is a left adjoint functor and this makes the groupoid version of S-vK. somehow trivial. (or at least it hides all the difficulty under the rug) $\endgroup$ Commented Sep 19, 2020 at 21:35
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1$\begingroup$ @curious: you do not get Seifert-van Kampen this way. What you get is that the fundamental groupoid functor preserves homotopy pushouts, but Seifert-van Kampen is a genuinely point-set result telling you when a point-set pushout is "close enough" to a homotopy pushout that the fundamental groupoid is what it should be. $\endgroup$ Commented Sep 19, 2020 at 21:49
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2$\begingroup$ @curious: you're working with $\infty$-categories and adjunctions between them and in $\infty$-category land you only have $\infty$-limits and colimits and those are what right adjoints resp. left adjoints preserve. $\endgroup$ Commented Sep 19, 2020 at 21:54
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1$\begingroup$ @QiaochuYuan Ah yes, thank you! $\endgroup$ Commented Sep 19, 2020 at 21:55