nLab Quillen negation

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Contents

Idea

(…)

Definition

Definition

(weak orthogonal class/Quillen negation)
Given a class MβŠ‚Mor(π’ž)M \;\subset\; Mor(\mathcal{C}) of morphisms in a category π’ž\mathcal{C}, its

  • left weak orthogonal class or left Quillen negation

    M β§„βŠ‚Mor(π’ž)\multiscripts{^⧄}{M}{} \;\subset\; Mor(\mathcal{C})

    (also denoted MProjM Proj or similar, for MM-projective morphisms)

or

  • right weak orthogonal class or right Quillen negation

    M β§„βŠ‚Mor(π’ž)\multiscripts{}{M}{^⧄} \;\subset\; Mor(\mathcal{C})

    (also denoted MInjM Inj or similar, for MM-injective morphisms)

is the class of morphisms in π’ž\mathcal{C} which have the left, respectively right, lifting property with respect to each morphism in the class MM:

M ⧄≔{p∈Mor(π’ž)|βˆ€i∈Mi⧄p}, ⧄M≔{i∈Mor(π’ž)|βˆ€p∈Mi⧄p}. 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 ⧄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{}^{⧄}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

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

Examples

Example

In the category Set of sets, the right Quillen negation of the β€œsimplest non-surjection” βˆ…β†’*\varnothing \to \ast (the unique map from the empty set to the singleton set) is the class of surjections:

{βˆ…β†’*} ⧄={f|fis surjective}. \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}β†’*\{x_1,x_2\} \to \ast (the β€œsimplest non-injection”) are both precisely the class of injections,

⧄{{x 1,x 2}⟢*}={{x 1,x 2}⟢*} ⧄={f|f is injective }. {}^{⧄} \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:

Last revised on September 8, 2022 at 08:14:17. See the history of this page for a list of all contributions to it.