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Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{Cat}(QX,Y)=\operatorname{Pseudo}(X,Y).$$

For various reasons, it is often more convenient to work with normalized pseudofunctors, that is, pseudofunctors preserving units strictly. Steve Lack and Simona Paoli gave a construction of such a comonad in the case $n=2$ implicitly in [2].

Is there any variant of Garner's construction, hopefully as part of a cofibrantly generated algebraic weak factorization system adapted to the folk model structure on strict $n$-categories that corepresents the normalized pseudofunctors in the same way?

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  • $\begingroup$ To which construction in the Lack--Paoli paper do you refer? There is a standard construction of the normal pseudofunctor classifier of a 2-category/bicategory (also called "normal strictification"), but I don't see it in that paper. $\endgroup$ – Alexander Campbell Dec 11 '18 at 13:53
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$\require{AMScd}$Notation: for each $n \geq 0$, let $\mathbf{2}_n$ denote the free-living $n$-cell, and let $\partial\mathbf{2}_n$ denote its boundary. Let $n$-Cat denote the category of (strict) $n$-categories and (strict) $n$-functors.

Recall (see e.g. Section 7.2 of Garner's `Understanding the small object argument') that the "pseudofunctor classifier" (or "strictification") comonad on the category 2-Cat is the cofibrant replacement comonad of the awfs (= algebraic weak factorisation system) on 2-Cat generated (via the algebraic small object argument) by the set of 2-functors $$\{\partial\mathbf{2}_0 \to \mathbf{2}_0, \partial\mathbf{2}_1 \to \mathbf{2}_1, \partial\mathbf{2}_2 \to \mathbf{2}_2, \partial\mathbf{2}_3 \to \mathbf{2}_2\}.$$

One can similarly obtain the "normal pseudofunctor classifier" (or "normal strictification") comonad on 2-Cat as the cofibrant replacement comonad for the awfs on 2-Cat generated by a certain category $\mathcal{J}_2$ of 2-functors (i.e. a subcategory of the arrow category of 2-Cat). The objects of this category $\mathcal{J}_2$ are the 2-functors listed above, together with the identity 2-functor $\mathbf{2}_0 \to \mathbf{2}_0$; the only non-identity morphism in $\mathcal{J}_2$ is the unique morphism in the arrow category of 2-Cat from $\partial\mathbf{2}_1 \to \mathbf{2}_1$ to $\mathbf{2}_0 \to \mathbf{2}_0$, i.e. the following commutative square of 2-functors. \begin{CD} \partial\mathbf{2}_1 \ @>>> \mathbf{2}_0\\ @V V V @VV V\\ \mathbf{2}_1 @>>> \mathbf{2}_0 \end{CD} (Intuitively, the effect of this change to the algebraic small object argument is that a cell is attached only for each "non-degenerate" lifting problem.)

The natural generalisation of this construction would be to define the "normal pseudofunctor classifier" comonad on $n$-Cat to be the cofibrant replacement comonad for the awfs on $n$-Cat generated by the category $\mathcal{J}_n$ of $n$-functors whose objects are the following "boundary inclusions" and identity $n$-functors: $$\{\partial\mathbf{2}_0 \to \mathbf{2}_0, \ldots, \partial\mathbf{2}_n \to \mathbf{2}_n, \partial\mathbf{2}_{n+1} \to \mathbf{2}_n\}\cup\{\mathbf{2}_0 \to \mathbf{2}_0, \ldots, \mathbf{2}_{n-1} \to \mathbf{2}_{n-1}\},$$ and whose only non-identity morphisms are the commutative squares \begin{CD} \partial\mathbf{2}_{k+1} \ @>>> \mathbf{2}_k\\ @V V V @VV V\\ \mathbf{2}_{k+1} @>>> \mathbf{2}_k \end{CD} (where the bottom $n$-functor picks out the identity $(k+1)$-cell of the non-identity $k$-cell of $\mathbf{2}_k$) for each $0 < k < n$. If we let $Q$ denote the comonad so constructed, this generalisation suggests that a normal pseudofunctor between $n$-categories $A \rightsquigarrow B$ is a strict $n$-functor $QA \to B$.

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  • $\begingroup$ Haha, Brilliant! I e-mailed you the same question earlier today, and this answer does not disappoint. It's really close to what I tried (using a subcat instead of subset of arrows), but yours works! Very nice and clean! $\endgroup$ – Harry Gindi Dec 11 '18 at 15:00

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