nLab Diers spectrum

Idea

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Definition

Suppose $U\colon A\to B$ is a functor that has a left multiadjoint.

The Diers spectrum of $U$ is the functor $B^{op}\to Set$ that sends an object $X\in B$ to the set $I$ that indexes the value $(g_i\colon B\to U(A_i))_{i\in I}$ of the left multiadjoint of $U$ at $X$. A morphism $f\colon X\to X'$ is sent to the induced map $I'\to I$ that assigns to $i'\in I'$ the unique $i\in I$ such that $g_{i'}\circ f\colon X\to U(A'_i)$ factors through $g_i$.

Example

The category of integral domains and injective homomorphisms is a multireflexive subcategory of the category of commutative rings and ring homomorphisms.

The associated Diers spectrum functor is a functor

$$CRing^{op} \to Set$$

that sends a commutative ring to its prime spectrum.

Example

The category of reduced rings and injective homomorphisms is a multireflexive subcategory of the category of commutative rings and ring homomorphisms.

The associated Diers spectrum functor is a functor

$$CRing^{op} \to Set$$

that sends a commutative ring to its poset of radical ideals. This is precisely the underlying poset of the localic Zariski spectrum.

Reference

• Yves Diers, Some spectra relative to functors. Journal of Pure and Applied Algebra 22:1 (1981), 57–74. doi

• Axel Osmond: On Diers theory of Spectrum I: Stable functors and right multi-adjoints, arXiv.

• Axel Osmond: On Diers theory of Spectrum II Geometries and dualities, arXiv.