# Comonad for normalized pseudofunctors for strict higher categories

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?

• 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. – Alexander Campbell Dec 11 '18 at 13:53

$$\require{AMScd}$$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 $$QA \to B$$.

• 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! – Harry Gindi Dec 11 '18 at 15:00