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It pertains to the classical literature on the subject (see e.g. here) the fact that an accessible 2-monad $T$ on a 2-category $\cal K$ induces an adjunction $$ \mathbf{Strict}\text{-}T\text{-}\textbf{Alg}\leftrightarrows \mathbf{Pseudo}\text{-}T\text{-}\textbf{Alg} $$ between strict $T$-algebras and pseudo $T$-algebras. This means that for every morphism $(A,a)\to (B,b)$ of $T$-algebras where the relevant diagrams commute up to invertible 2-cells, there is a new $T$-algebra $(A^\text{s}, a^\text{s})$ with a strict morphism $(A^\text{s}, a^\text{s}) \to (B,b)$.

What does this construction become when $T : {\bf Cat}\to {\bf Cat}$ is the 2-monad that sends a small category $A$ into the category $A^\textbf{2}$ of its arrows (for sure a finitary monad on a locally finitely presentable category)?

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  • $\begingroup$ Three irresponsible blind guesses: either$${\bf Cat}^\textbf{2}\leftrightarrows{\bf Cat}/\textbf{2}$$or$${\bf Cat}^\textbf{2}\leftrightarrows{\bf Cat}//\textbf{2}$$or$${\bf Cat}/\textbf{2}\leftrightarrows{\bf Cat}//\textbf{2}$$(here "//" stands for hypercomma, a. k. a. lax slice) $\endgroup$ May 8 '17 at 8:46
  • $\begingroup$ A single naive question: are you saying that ${\bf Cat}^2\cong {\bf Cat}/2$? I've never noticed this. $\endgroup$
    – fosco
    May 8 '17 at 8:58
  • $\begingroup$ Wait, this is not the only problem. What do these three adjunctions denote? No one of the categories in your guess is the category of algebras for the squaring monad :-) $\endgroup$
    – fosco
    May 8 '17 at 9:02
  • $\begingroup$ Re first - lhs embeds into rhs as fibrations over $\bf2$. Re second - I believe all three admit adjunctions to $\bf Cat$ inducing the $(\_)^{\bf2}$ monad. $\endgroup$ May 8 '17 at 9:21
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    $\begingroup$ Just an à propos: obviously "codescent" shall be banned, and replaced by "ascent" :D $\endgroup$ May 10 '17 at 7:41
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A strict algebra of the squaring monad is precisely a strict factorization system. A normal pseudo-algebra (unit holds strictly) is precisely an orthogonal factorization system. See here. The pseudo-algebras will differ by first replacing a morphism by an isomorphic morphism before factoring it.

So that's what the pseudo-algebras look like. You can read off from this a description of the strictified monad.

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