nLab Quillen negation

Idea

(…)

Definition

Definition

(weak orthogonal class/Quillen negation)
Given a class $M \;\subset\; Mor(\mathcal{C})$ of morphisms in a category $\mathcal{C}$, its

• left weak orthogonal class or left Quillen negation

  $\multiscripts{^⧄}{M}{} \;\subset\; Mor(\mathcal{C})$

  (also denoted $M Proj$ or similar, for $M$-projective morphisms)

or

• right weak orthogonal class or right Quillen negation

  $\multiscripts{}{M}{^⧄} \;\subset\; Mor(\mathcal{C})$

  (also denoted $M Inj$ or similar, for $M$-injective morphisms)

is the class of morphisms in $\mathcal{C}$ which have the left, respectively right, lifting property with respect to each morphism in the class $M$:

$$M^{⧄} \;\coloneqq\; \Big\{ p \,\in\, Mor(\mathcal{C}) \;\big\vert\; \underset{i \in M}{\forall} \; i \,⧄\, p \Big\}, \;\;\;\;\; {}^{⧄}M \;\coloneqq\; \Big\{ i \,\in\, Mor(\mathcal{C}) \;\big\vert\; \underset{p \in M}{\forall} \; i \,⧄\, p \Big\} \,.$$

Properties

Proposition

(closure properties)
A right weak-orthogonal class $C^{⧄}$ is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) and composition of morphisms, and contains all isomorphisms of C.

Meanwhile, a left weak-orthogonal class ${}^{⧄}M$ is closed under retracts, pushout, (small) coproduct and transfinite composition (filtered colimits) of morphisms (whenever they exist in the category), and also contains all isomorphisms.

This proof is straightforward, see for instance here at injective or projective morphism.

Remark

It is clear that

$$C \;\subset\; \big({}^{⧄}M\big)^{⧄} \,, \;\;\; C \;\subset\; \multiscripts{^⧄}{\big(M^{⧄}\big)}{} \,.$$

Related entries

• lifting property

• separation axioms in terms of lifting properties

Examples

Example

In the category Set of sets, the right Quillen negation of the “simplest non-surjection” $\varnothing \to \ast$ (the unique map from the empty set to the singleton set) is the class of surjections:

$$\big\{ \varnothing \to \ast \big\}^{⧄} \;=\; \big\{ f \;\big\vert\; f \; \text{is surjective} \big\} \,.$$

The left and right Quillen negations of $\{x_1,x_2\} \to \ast$ (the “simplest non-injection”) are both precisely the class of injections,

$${}^{⧄} \big\{ \{x_1,x_2\} \longrightarrow \ast \big\} \;=\; \big\{ \{x_1,x_2\} \longrightarrow \ast \big\}^{⧄} \;=\; \big\{ f \;\big\vert\; f \; \text{ is injective } \big\} \,.$$

References

Left/right weak orthogonal classes are discussed, for instance, in any text on model category-theory, see the references there.

Discussion with an eye towards separation axioms in terms of lifting properties and amplifying the logical aspect of conceptual negation:

• Misha Gavrilovich, Remarks on Shelah’s classification theory and Quillen’s negation (arXiv:2010.08791)