Introduction E9F5FC Understandable FFFFFF Questions FFFFC0 Notes EEEEEE Software 
Kerodon Definition 1.1.1.2 (The Simplex Category). Let {$Δ$} denote the category whose objects are sets of the form {$[n]$} (where {$n$} is a nonnegative integer), where a morphism {$[m]$} to {$[n]$} is a nondecreasing function {$α:[m]→[n]$} (that is, a function α which satisfies the condition {$α(i)≤α(j)$} whenever {$i≤j$}). We refer to {$Δ$} as the simplex category. An object {$[n]$} is an nsimplex, or alternatively, a rows of cells of length n+1. Given a morphism from {$[m]$} to {$[n]$}, all elements of {$[m]$} must be used exactly once (thus they are interpreted as cells in space) whereas an element of {$[n]$} may be used multiple times or not at all (thus they are interpreted as labels). Temporarily, we can think of the cells of {$[m]$} as expressing the observed's time, whereas the labels from {$[n]$} are expressing the observer's time. A morphism from {$[m]$} to {$[n]$} indicates how an msimplex may grow and arise in n+1 steps, or alternatively, how a row of cells of length m+1 may be filled by a row of increasing numbers taken from {$[n]$}. Note that not all labels need to be used. Thus each morphism from {$[m1]$} to {$[n1]$} can be identified with a polynomial term of degree {$m$} given by weights taken from {$[n1]$}, which is to say, {$x_{i_1}^{j_1}x_{i_2}^{j_2}\dots x_{i_n}^{j_n}$} where {$\sum_{k=1}^{n}j_k=m$}. The homogenous symmetric function {$h_m(x_1,x_2,\dots ,x_n)$} is the generating function for the set of all morphisms from {$[m1]$} to {$[n1]$}. Composition of morphisms leads to a redefinition of the partition based on substitution of the labels. The labels may be remapped, but parts can only be merged, not refined. Readings 
SimplexCategoryNaujausi pakeitimai 
Puslapis paskutinį kartą pakeistas 2019 balandžio 16 d., 14:51
