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Exact true-triangle and Waldhausen determinant one-type input

PDF p. 9: Actual low-weight projections and determinant normalization

Statement

Assume AC and fixed universe bounds. For a small Waldhausen category W a determinant functor to a Picard groupoid P consists of a functor on weak equivalences and additivity isomorphisms for actual cofibration sequences, satisfying weak-equivalence naturality, staircase associativity and commutativity. It factors essentially uniquely through the universal Picard groupoid V(W), which represents the 1-type of KW(W). Hence it gives an actual map of connective K-space 1-types to the nerve of P, natural in exact functors and compatible with object/automorphism classes. For bounded finite-projective complexes, the Knudsen–Mumford alternating graded determinant is such a functor. Its triangle coherence is for true/enhanced triangles. It is not a claim of natural additivity under every morphism of ordinary distinguished-triangle diagrams.

External input

The statements below are precise established inputs. Their constructions are used at the Waldhausen/cofiber-sequence and Picard-one-type level. No general determinant functor on arbitrary diagrams in an ordinary triangulated category, or unrestricted higher categorical enhancement of Knudsen–Mumford, is imported.

Facts & Assumptions

Given: The stated model hypotheses, AC and fixed universe bounds.

Goal: Supply the actual determinant map required by the consumers, with exactly the recorded coherence.

[F1]

The precise KM and MTW source records above.

[F2]

The BGT record above supplies the canonical enhanced/connective/nonconnective comparison under its explicit hypotheses.

Proof

1.1F1

Apply the Waldhausen branches of MTW Definition1.2.3, Theorems1.5.4 and1.6.3. Definition1.2.9 identifies the supplied determinant functor with a symmetric monoidal functor out of V(W), and the 1-type comparison supplies the actual K-space map with its object and weak-equivalence maps. The finite Picard target needs this coherence, rather than any replacement of it by bare equality of K0,K1 groups.

2.1F1step 1.1

KM Definition4 provides the weak-equivalence naturality and associativity on actual true short-exact complex diagrams; its restriction to bounded finite-projective complexes is explicitly the MTW Example1.2.7. The graded-line symmetry is retained, matching the commutativity axiom and its sign. KM pullback isomorphisms include composition compatibility. Thus these data are natural under tensor extension of bounded projectives and under restriction.

3.1F2step 1.1∎

Apply BGT to the bounded-projective Waldhausen model when transporting the map to the specified affine enhanced Perf model. Idempotent completeness supplies its missing degree-zero hypothesis in Remark9.33, in addition to positive-degree agreement. The canonical comparison is therefore an equivalence on connective covers, and the determinant map transports across that actual equivalence. This imports the exact model bridge, not a new enhancement inferred from KM’s ordinary statement or local Postnikov calculations.

Depends on

Used by · 1

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External dependencies

Knudsen–Mumford, The projectivity of the moduli space of stable curves I: Preliminaries on det and Div · graded invertible sheaves pp20–21; alternating determinant p31; Definition2 p37 and remark p38; Definition4 pp40–42 including footnote p41; Theorem2 p42; warning and Proposition7 p44

Extended determinant is defined on perfect complexes with additivity for true triangles, represented in the derived category of actual short-exact complex diagrams, with naturality for their isomorphisms, the true-triangle-of-triangles axiom, and derived-pullback composition compatibility. The source expressly does not assert functorial additivity for all ordinary triangle diagrams on arbitrary schemes.

Hypotheses: Arbitrary schemes and locally perfect complexes for Theorem2. True triangles and isomorphisms are those of VT(A), not all commuting diagrams in D(A). Additional reducedness hypothesis occurs in Proposition7 for an ordinary-triangle extension and is not silently imposed on Theorem2.

Reading limits: This assignment visually read the cached primary KM scans of printed pp.20–21,31,37–38,40–44, including the true-triangle category, all Definition4 axioms and its p41 footnote, Theorem2, the perfect-cohomology corollary and p44 warning. The remote journal URL was unavailable. Theorem2 and the omitted Proposition4 proof remain established external inputs; no reconstruction or all-higher enhancement of KM is claimed.

Fernando Muro, Andrew Tonks and Malte Witte, On determinant functors and K-theory, Publicacions Matematiques59(2015),137–233 · Definition1.2.3 and axioms pp145–146; Example1.2.7 p147; Definition1.2.9 pp147–148; Theorem1.5.4 pp161–162; section1.6 Lemma1.6.1 p166 and Theorem1.6.3 p167; natural construction statement surrounding Theorem4.3.4 p212

A determinant functor on a Waldhausen category is a functor on weak equivalences with additivity for actual cofibration sequences, naturality under weak-equivalence diagrams, staircase associativity and commutativity. Its universal Picard target exists and models the actual Waldhausen K-spectrum 1-type. Example1.2.7 explicitly includes the bounded-complex extension of the graded vector-bundle determinant.

Hypotheses: Small Waldhausen category up to fixed universes. Picard groupoid target; retain its graded symmetry. Theorem1.6.3 gives the actual Waldhausen branch, not a substitution of arbitrary triangulated K-theory. Exact functors preserve the data and the universal construction is natural.

Reading limits: Read extracted text of the author-hosted 2015 journal PDF, printed pp.145–148,161–162,166–167,212: the complete named definitions, both determinant axioms, Example1.2.7, universal factorization, the named theorem statements, and the p212 naturality/2-functoriality passage. The long universal-construction and homotopy-model proofs are imported, not reconstructed. Only the Waldhausen branch is applied.

Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory · Lemma7.11 and Corollary7.12 pp48–49; Remark9.33 p64; compare completed enhanced-comparison supplier

The Waldhausen-to-infinity K comparison is natural for weakly exact functors under DHKS saturation and factorization. The canonical connective-to-nonconnective map agrees in positive degrees; its degree-zero comparison is an isomorphism under idempotent completeness. Thus its connective truncation is an equivalence for Perf(A).

Hypotheses: DHKS-saturated Waldhausen category admitting factorization for the model comparison. Idempotent-complete associated stable infinity-category is needed for degree zero; Perf(A) satisfies it.

Reading limits: Read arXiv:1001.2282v4 (5 February 2013), pp.48–49 in full for Lemma7.11/Corollary7.12 and their proofs, and p64 Proposition9.32/proof and Remark9.33. The present application explicitly uses idempotent completeness for degree zero as well as positive-degree agreement. The corepresentability and delooping foundations remain imported.

Sources