Categories and morphisms

Overview

• Topological categories and functors

Related notes: Categorical functors

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Basic definitions

Definition: Category

A category $\mathsf{C}$ consists of four pieces of data:

1. A collection $\mathsf{Ob}(\mathsf{C})$ of “objects”.
2. For each pair of objects $X,Y\in \mathsf{Ob}(\mathsf{C})$, a set $\mathsf{C}(X,Y)={\mathsf{Hom}}_{\mathsf{C}}(X,Y)$ of morphisms from $X$ to $Y$. A generic element $f\in {\mathsf{Hom}}_{\mathsf{C}}(X,Y)$ is typically written $f:X\to Y$.
3. For each triple of objects $X,Y,Z\in \mathsf{Ob}(\mathsf{C})$, we have a composition map ${\mathsf{Hom}}_{\mathsf{C}}(Y,Z)\times {\mathsf{Hom}}_{\mathsf{C}}(X,Y)\to {\mathsf{Hom}}_{\mathsf{C}}(X,Z)$ defined by $(g,f)\mapsto g\circ f$.
4. For each $X\in \mathsf{Ob}(\mathsf{C})$, we specify an identity morphism $1_X\in {\mathsf{Hom}}_{\mathsf{C}}(X,X)$.

This data is required to satisfy the following properties:

• Associativity of composition. For each four-tuple of objects $X,Y,Z,W\in \mathsf{Ob}(\mathsf{C})$, if $f:Z\to W$, $g:Y\to Z$, and $h:X\to Y$ are composable morphisms, then $(f\circ g)\circ h=f\circ (g\circ h).$
• Composition with identity. If $f:X\to Y$ is a morphism, then $f\circ 1_X=1_Y\circ f=f.$

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Isomorphisms

Definition: Isomorphism

Let $\mathsf{C}$ be a category. Then a morphism $f:X\to Y$ in $\mathsf{C}(X,Y)$ is an isomorphism if there exists a morphism $g:Y\to X$ in $\mathsf{C}(Y,X)$ such that $g\circ f=1_X\in \mathsf{C}(X,X)f\circ g=1_Y\in \mathsf{C}(Y,Y).$

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Examples

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Notes

\mathsf{Ob}(\mathsf C)

\mathsf {Hom}_{\mathsf C}(X, Y)