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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 n-simplex, 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 m-simplex 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 {$[m-1]$} to {$[n-1]$} can be identified with a polynomial term of degree {$m$} given by weights taken from {$[n-1]$}, 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 {$[m-1]$} to {$[n-1]$}.

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

• nLab: Simplex Category
• 0001 The Language of ∞-Categories