nLab
Diers spectrum
Idea
[…]

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 .

Last revised on April 8, 2021 at 14:36:44.
See the history of this page for a list of all contributions to it.