Category theory MOC

Category semigroup

Let $𝑅$ be a commutative ring and $𝖢$ be a small category. The category rng $K 𝖢$ is a $K$ -semigroup constructed from the free module $K (𝖢 1)$ . #m/def/cat This is a generalization of the monoid ring in light of monoids as categories. In the case $O b (𝖢)$ is finite, this construction gives an extension ring of $K$ called the category ring.

Construction

We begin with the free module $𝑅 (𝖢)$ taking the objects as identities convention, and linearly extend the following product for $𝑓, 𝑔 \in 𝖢$

𝑔 \cdot 𝑓 = {0 c o d 𝑓 \neq d o m 𝑔 𝑓 \circ 𝑔 c o d 𝑓 = d o m 𝑔 .

If $O b (𝖢)$ is finite, then this forms an $K$ -monoid with an identity given by

1 = \sum 𝑥 \in O b (𝖢) 1 𝑥 .

Properties

Module over a category ring

Special case

Path algebra

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