Mapping cylinders

Map-induced persistence (Kernel, image, cokernel persistence) is defined for an inclusion \(L \hookrightarrow K\). When the simplicial map you actually care about is not an inclusion – collapses, merges, glueings, two different filtrations on overlapping vertex sets – the standard fix is to replace the map with its mapping cylinder, then run KICR on the resulting inclusion.

The construction

Given two filtrations on overlapping vertex sets, the mapping cylinder is

\[ \text{Cyl}(f) = \big(L \times \{0\}\big) \;\cup\; \big(K \times \{1\}\big) \]

glued by the cylinder cells. In Oineus this is realized as a product filtration: each cell in the cylinder is a product of a cell from one of the input filtrations with an auxiliary vertex (v_top or v_bottom).

API

import oineus as oin

# fil_K: the "big" filtration. fil_L: the inclusion source.
v_top    = oin.Simplex([n_K + n_L],     0.0)   # auxiliary vertex at t=0
v_bottom = oin.Simplex([n_K + n_L + 1], 0.0)   # auxiliary vertex at t=1

fil_cyl = oin.mapping_cylinder(fil_L, fil_K, v_top, v_bottom)

mapping_cylinder(fil_domain, fil_codomain, v_domain, v_codomain) returns a oineus.ProdFiltration of product cells. The two auxiliary vertices need IDs that don’t collide with the vertices already in either filtration. Their values default to the filtration’s \(-\infty\) sentinel (so they enter the filtration before any real cell); pass v_domain_value=... / v_codomain_value=... to override.

When to use this

For the common case of KICR on a cylinder, the wrapper oineus.compute_ker_cok_reduction_cyl() builds the cylinder, the inclusion, and the KICR object in one call:

kicr_prod = oin.compute_ker_cok_reduction_cyl(fil_2, fil_3)

kicr_prod.kernel_diagrams().in_dimension(0)
kicr_prod.cokernel_diagrams().in_dimension(0)

It returns a oineus.KerImCokReducedProd (the product-cell analogue of oineus.KerImCokReduced; same accessors).

Building blocks

If you need to assemble a cylinder by hand:

  • oineus.mapping_cylinder() – the cylinder itself.

  • oineus.multiply_filtration() – \(\text{fil} \times \{\sigma\}\), i.e. multiply every cell in fil by the auxiliary cell. This is how you build the inclusion-as-subcomplex of the cylinder.

  • oineus.min_filtration() – the pointwise minimum of two filtrations on a shared vertex set. Useful when the “two filtrations” picture starts with two unrelated functions on the same complex.

See also