Visualization reference

visualize(object; kind=:auto, ...) builds a mathematical specification and renders it through an explicitly loaded backend. visual_spec(object; ...) returns that inspectable specification without rendering. The latter and its accessors are available through TamerOp.Advanced. Use available_visuals to list the supported kinds for the actual object, and check_visual_request to inspect its accepted keyword and cost contract.

Resolutions and presentation incidence

These overloads accept DerivedFunctors.ProjectiveResolution, DerivedFunctors.InjectiveResolution, IndicatorResolutions.UpsetResolutionResult, IndicatorResolutions.DownsetResolutionResult, and the workflow Results.ResolutionResult wrapping either derived resolution. They consume stored terms, not just a bare matrix of multiplicities, and never construct or extend a resolution. The resolution guide works through a diamond example with known Betti and Bass tables.

Degree means homological degree for projectives and cohomological degree for injectives. Degree zero selects the (co)augmentation when vertex is given. A positive degree k selects P_k → P_(k-1) or I^(k-1) → I^k respectively. summand is a one-based index in the selected term's stored summand order; it highlights the projective source column or injective target row. An empty term has no selectable summand. vertex is a finite-poset ID, not an ambient parameter. Its active rows and columns retain their global summand IDs. If matrix_limit crops the coefficient panel, the selected row or column is retained within that limit. Metadata records the displayed global indices.

A default table reports unverified multiplicities. verify=true checks stored principal bases, agreement of differential coefficients with stored maps, augmentation naturality, augmented equations, and vertexwise exactness through the computed prefix. It tests each cover/hull for minimality and the terminal kernel/cokernel for completion. A valid nonminimal resolution is viewable and labelled :nonminimal; malformed or inexact prefixes are rejected. A nonzero terminal kernel/cokernel gives :truncated. Missing higher degrees are never padded with zero rows. Exact fields use exact coefficients and rank; RealField uses its declared equality tolerances and numerical-rank backend. Verification is fresh and does not trust cached minimality summaries.

grades supplies one finite real coordinate pair per finite vertex. Its coordinatewise order must agree with the finite order. No coordinates are inferred from a Hasse diagram or an encoding classifier's representative points. Exact supplied grades remain in metadata; plotting converts them to floating coordinates. Distinct grades that collide in those display coordinates are rejected; use the exact table instead. A grade plot makes no claim about an ambient multigraded free resolution. A multiplicity label counts coincident summands at a vertex; the combined resolution view highlights the selected summand's vertex.

support_sheets=true shows separately labelled principal supports in the selected term. matrix_limit[1] bounds the number of these panels; an explicitly selected summand is also retained. Panel separation is schematic and does not supply an extra parameter or a decomposition of the resolved module.

Cokernel presentations and kernel copresentations

IndicatorTypes.UpsetPresentation and DownsetCopresentation support :presentation_incidence as their default. They accept the resolution selection keywords except verify. Degree 0 selects generators/cogenerators; degree 1 selects relations/corelations. Both show the same single presentation map. Supports must be principal on the supplied finite poset. A projective presentation is displayed as a cokernel; the downset copresentation is a kernel. Neither is the fringe image shown by :presentation_inspector. No minimality or subsequent resolution is inferred.

All displayed coefficient matrices use target rows and source columns. This transposes the stored UpsetPresentation.delta convention. Labels identify the actual summands and, when supplied, their grades. A dagger marks an order-forced zero; an ordinary zero is allowed by order. At a selected vertex, the active matrix determines the reported cokernel/kernel dimension.

Supplied graded basis changes

On a selected differential or presentation, use basis_change=(; source=S, target=T). The square matrices must use the same coefficient field and stored source/target summand counts. Their coefficients must obey the principal-support order; both must be invertible. The viewer computes B = inverse(T)*A*S and checks A*S = T*B. It retains both matrices, the changes and their inverses, and the equation result in metadata.

This certifies the selected map in different graded coordinates, not a minimalization or decomposition algorithm. It does not transform adjacent maps or mutate the original object. The augmentation's target is not generally a principal sum, so resolution basis changes require a positive selected degree.

Specification data and costs

visual_metadata(spec) retains counts, stored degrees, computed length, category, field, direction, verification status, selection and supplied grades. Incidence views additionally retain coefficient_matrix, stalk_matrix, incidence, basis_change, and displayed support-sheet indices/counts. A presentation stalk also supplies represented_dimension. Matrix panels retain the full selected matrix, its original shape and displayed row/column ranges. These arrays are copied; editing them does not change the input resolution.

Default tables count generators without elimination. A selected injective coefficient matrix is recovered from its recorded stalk maps. Explicit verification traverses the whole stored prefix and performs field-aware ranks; a basis change performs inversions and equation checks. Matrix display limits do not limit those mathematical checks or discard the exact selected matrix. Rendering receives only prepared data and performs no resolution algebra. These views use argument-based selections with CairoMakie or WGLMakie; they do not supply live resolution inspector controls.