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While reading about accessible categories in Locally Presentable and Accessible Categories I came accross the notion of $\lambda$-pure subobjects, which seem to be important while characterising accessible subcategories of an accessible category.

What is the intuition behind considering this particular class of subobjects? Is it just because it is the smallest class of subobjects containing split subobjects which is closed under $\lambda$-directed colimits?

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Think of the category of structures in some signature $\Sigma$. The functor represented by a finitely presentable object $F$, say, corresponds to some term in the language generated by $\Sigma$. For example, suppose that $\Sigma$ is the signature for fields, and $F = \mathbb{Q}[x]/(f(x))$ for some irreducible polynomial $f(x)$. Then a morphism $F \to K$ corresponds to an element $\alpha \in K$ satisfying the equation $f(\alpha) = 0$. Let $I$ be the initial structure in this language. If $\phi: K \to L$ is a morphism, then to say that $\phi$ is $\omega$-pure with respect to the morphism $I \to F$ is to say the following: if $f$ has a root in $L$, then it already has a root in $K$. So this says that the morphism $K \to L$ reflects satisfaction of the formula $\exists x f(x) = 0$.

Using morphisms $F_1 \to F_2$ where $F_1$ is not initial, we can do something similar for formulas with parameters in $K$. For example, suppose that $F_1$ is the field $\mathbb{Q}[x] / f(x)$ and $F_2$ is the field $F_1[y] / g(x,y)$ for some polynomial $g$. Then $\omega$-purity with respect to the inclusion $F_1 \to F_2$ says that if $\alpha$ is a root of $f$ in $K$, and if there is a root of $g(\alpha,y)$ in $L$, then there is already a root of $g(\alpha,y)$ in $K$.

So basically, purity of a morphism $K \to L$ means that certain types of existence statements, if satisfied in $L$, are already satisfied in $K$. This says that there is a "reflection principle" for the inclusion $K \to L$: if you can solve an equation in $L$, then you can already solve it in $K$. This doesn't say that $K \to L$ is an elementary embedding, but it does say that the map is elementary with respect to some $\Sigma_1$ formulas.

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  • $\begingroup$ See also Proposition 5.15 in the mentioned book by Adamek-Rosicky. (I wish the authors had made a reference in chapter 2 to this section, because in chapter 2 most of the theory of pure morphisms is developed without giving any specific characterizations in specific categories.) $\endgroup$
    – HeinrichD
    Commented Dec 16, 2016 at 9:01
  • $\begingroup$ I think I saw this explanation on nLab as well. But this idea seems to make sense only when talking about some algebraic category. Do you know how this translates into other categories, for example the category of presheafs over a small category? Can we say something about pure morphisms in a general locally presentable category? $\endgroup$
    – Arun Kumar
    Commented Dec 16, 2016 at 10:36
  • $\begingroup$ This description holds for categories of structures (with relations and sorts also allowed). Now one can use Prop 2.32 and Ex. 1.41 to apply this to presheaf categories. I assume that you already know Cor. 2.30 that $\lambda$-pure morphisms in a locally $\lambda$-pres. category are the $\lambda$-directed colimits of split monos? $\endgroup$
    – HeinrichD
    Commented Dec 17, 2016 at 7:14
  • $\begingroup$ @ArunKumar presheaf categories are examples of categories of structures, with one sort for each object, no relation symbols, and one unary function symbol for each morphism. In general, since a $\lambda$-accessible category embeds fully faithfully into the category of presheaves on the $\lambda$-presentable objects, you can think of it as a category of structures in at least one way. $\endgroup$
    – Tim Campion
    Commented Dec 19, 2016 at 1:50
  • $\begingroup$ Thanks! I neglected the relational structures example on my first reading. I'll read that in more detail now. $\endgroup$
    – Arun Kumar
    Commented Dec 19, 2016 at 14:49

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