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Many invariants are given by topological invariants composed with the nerve functor from (small) categories to topological spaces. For example, you can talk of $\pi_0(C)$ of a category $C$ and it is exactly what you might guess, the set of connected components. But also higher homotopy groups are defined. This article by Tom Leinster studies the Euler characteristic of a finite category.

Also note that every property of categories can be considered as a functor $Cat \to 2$.

The center of a category provides a functor from $Cat$ (but only equivalences are allowed here) to the category of commutative monoids. This recovers, for example, $R$ from the category $R$-Mod. The automorphism class group of a category $C$ is the group of all equivalences $C \cong C$ up to isomorphism. This provides a functor $Cat \to Groups$.

An instance of decategorification is the functor $Cat \to Set$ which takes every (essentially small) category to the set of isomorphism classes of its objects.

There are certainly more invariants when you restrict to special categories. For example, Gabriel defined in his thesis the dimension of an abelian category, which has the property dim $R$-Mod = dim $R$ for noetherian $R$. But you can also define the global projective / injective dimension of an abelian category. Monoidal categories have an underlying monoid, which consists of the endomorphisms of the unit object. Every finite groupoid has a cardinality.

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Many invariants are given by topological invariants composed with the nerve functor from (small) categories to topological spaces. For example, you can talk of $\pi_0(C)$ of a category $C$ and it is exactly what you might guess, the set of connected components. But also higher homotopy groups are defined. This article by Tom Leinster studies the Euler characteristic of a finite category.

Also note that every property of categories can be considered as a functor $Cat \to 2$.

The center of a category provides a functor from $Cat$ (but only equivalences are allowed here) to the category of commutative monoids. This recovers, for example, $R$ from the category $R$-Mod. The automorphism class group of a category $C$ is the group of all equivalences $C \cong C$ up to isomorphism. This provides a functor $Cat \to Groups$.

An instance of decategorification is the functor $Cat \to Set$ which takes every (essentially small) category to the set of isomorphism classes of its objects.

There are certainly more invariants when you restrict to special categories. For example, Gabriel defined in his thesis the dimension of an abelian category, which has the property dim $R$-Mod = dim $R$ for noetherian $R$. But you can also define the global projective / injective dimension of an abelian category. Monoidal categories have an underlying monoid, which consists of the endomorphisms of the unit object.

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Many invariants are given by topological invariants composed with the nerve functor from (small) categories to topological spaces. For example, you can talk of $\pi_0(C)$ of a category $C$ and it is exactly what you might guess, the set of connected components. But also higher homotopy groups are defined. This article by Tom Leinster studies the Euler characteristic of a finite category.

Also note that every property of categories can be considered as a functor $Cat \to 2$.

The center of a category provides a functor from $Cat$ (but only equivalences are allowed here) to the category of commutative monoids. This recovers, for example, $R$ from the category $R$-Mod.

An instance of decategorification is the functor $Cat \to Set$ which takes every (essentially small) category to the set of isomorphism classes of its objects. As with

There are certainly more invariants when you restrict to special categories. For example, Gabriel defined in his thesis the last functordimension of an abelian category, which has the only morphisms in property dim $Cat$ R$-Mod = dim$R$for noetherian$R\$. But you can also define the global projective / injective dimension of an abelian category. Monoidal categories have an underlying monoid, which are allowed here are equivalencesconsists of the endomorphisms of the unit object.

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