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Homology of a Small Category with Functor Coefficients
by
Jing Wang
George Washington University
Coauthors: Jozef Przytycki

We first introduce the definition of homology of a small category with functor coefficients. Let C be a small category (i.e. Ob(C) forms a set), and F: C → R-Mod be a covariant functor from C to the category of modules over a commutative ring R. We call a sequence of objects and morphisms x₀ → x₁ → ...→ x_n an n-chain (formally n-chain in the nerve of the category).We define the chain group C_n to be direct sum of F(x₀) where the sum is taken over all n-chains x₀ → x₁ → ...→ x_n. The boundary operation ∂ is defined to be the alternating sum of face maps d₀, d₁, ..., d_n where for t ∈ F(x₀), d₀(t;x₀ → x₁ → ...→ x_n) = (F(x₀ → x₁)(t);x₁ → ...→ x_n) and for i > 0, d_i(t;x₀ → x₁ → ...→ x_n) = (t;x₀ → x₁ → ...→ x_i-1 → x_i+1 → ...→ x_n). A standard checking shows that ∂²=0. The yielded homology is called homology of category C with Coefficients in functor F.

In the case of a category where objects have a well-defined ßize" we may often build a chain complex in a simplified way. This is the case when we deal with the category of an abstract simplicial complex. Let K = (V; P) be an abstract simplicial complex with vertices V (which we order) and simplexes P. We consider K as a small category with Ob(K) = P and morphisms are inclusions. Let F be a covariant functor from K^op to the category of modules over a commutative ring R where K^op denotes the opposite category of K. We define the chain group C_n to be the direct sum of F(S) where he sum is taken over all n-dimensional simplexes S. The boundary operation ∂ is given by the alternating sum of face maps d₀, d₁, ..., d_n where if S = (v₀, v₁, ..., v_n) is an ordered simplex of dimension n then on F(S), d_i = F(S ⊃ (S-v_i)). Again we have ∂²=0. Thus we have a second definition of homology of an abstract simplicial complex category. It turns out that for an abstract simplicial complex category, these two definitions give the same homology groups. This is a known result for specialists. However, an elementary proof will be given here geared towards non-specialists.

Finally, we give another example which realizes Khovanov homology as homology of a simplex with coefficients in a specified Khovanov functor.

Date received: March 10, 2012

Copyright © 2012 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbek-14.