# Reference

## Simplicial Complexes

### Accessing

Hodge.numsimplicesFunction
numsimplices(sc[, k])

Return the number of k-dimensional simplices of sc.

If the parameter k is not given, return the total number of simplices including the empty face.

This function is a more efficient implementation of length ∘ simplices.

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## Cochains

### Construction

Hodge.CochainType
Cochain{R, n}

Represent the n-th group $C^n(K; R)$ of cochains over the ring R whose basespace is the simplicial complex $K$.

For constructing Cochains, see also the methods zero_cochain and indicator_cochain.

The elements of this type may be seem as functions from the n-simplices of K to R or as skew-symmetric n-tensors over the vertices of K. This second perspective follows the ideas from the paper:

• Jiang, X., Lim, L., Yao, Y. et al. Statistical ranking and combinatorial Hodge theory. Math. Program. 127, 203–244 (2011). https://doi.org/10.1007/s10107-010-0419-x
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Hodge.zero_cochainFunction
zero_cochain(R, K, n)

Construct an identically zero n-cochain over R and whose base space is K.

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Hodge.indicator_cochainFunction
indicator_cochain(R, K, σ)

Return the indicator function f of the n-simplex σ as a Cochain. That is, a cochain such that f(σ) = 1 and f(τ) = 0 for all other n-simplices of K.

Notice that, nevertheless, f is still skew-symmetric over permutations of σs indices.

If Κ does not contain σ, the returned cochain is identically zero.

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### Operators

Hodge.innerFunction
inner(ω, ξ)

Usual inner product between Cochains.

Warning

For complex Cochains, the conjugation is taken on the first entry.

This inner product sees a n-cochain as a free vector space over the (non-oriented) n-simplices of their base space. Formally,

$$$\sum_{\sigma \in \mathrm{simplices}(K,n)} \overline{f(σ)} g(σ).$$$
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