Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* ---- > [!definition] Definition. ([[group]]) > A **group** is a pair $(G, \cdot )$ consisting of a set $G$ and a [[binary operation]] $\cdot: G\times G \to G$ satisfying: > 1. *Identity.* $G$ has a $\cdot$-[[identity]], denoted $e_{G}$. Thus $G$ is a [[pointed set]]. > 2. *Inverses.* To every $g \in G$ corresponds a $\cdot$-[[inverse map|inverse]] $h$, $g \cdot h = e_{G}=h \cdot G$. > 3. *Associativity*. The operation $\cdot$ is [[associative]]. > > Groups are objects of [[category]] $\mathsf{Grp}$. The morphism set $\text{Hom}_{\mathsf{Grp}}(G,H)$ is the set of [[group homomorphism|group homomorphisms]] from $G$ to $H$. > > ^e1b4e3 > [!equivalence] Category-Theoretic Characterizations. > - A [[group]] $G$ is precisely a [[monoid]] where every element has an inverse, that is, a one-object [[category]] where every morphism is an [[isomorphism]]. Note that, cf. [[monoid#^note|this discussion]], a [[covariant functor|functor]] $G \to \mathsf{Grp}$ is precisely a [[group homomorphism]]. >- A [[group]] $G$ is precisely a [[groupoid]] with a single object ^equivalence > [!definition] Definition. (Group object) ^definition Let $\mathsf{C}$ be a [[category]] with [[categorical product|finite products]] (including a [[terminal object|final object]] $1$). A **group object** in $\mathsf{C}$ is an object $G$ of $\mathsf{C}$, together with morphisms $*:G \times G \to G$, $(\_)^{-1}: G \to G$, and $e:1 \to G$, such that - $*$ is associative: Create a table for different operations with group object like | Operation | For groups | For group objects | | ---------------------- | ---------------------------------- | ----------------- | | Associativity axiom | | | | Identity axiom | | | | Inverses axiom | | | | Homomorphism $\varphi$ | $\varphi(gh)=\varphi(g)\varphi(h)$ | (string diagram) | Group object examples: [[topological group]], [[Lie group]], measurable group... I think in [[global flow|one-parameter group action]] we want to reframe with group object actions (put in group action note). Maybe also in things like [[one-parameter semigroup]] but not sure. , one important wrinkle is that in Markov categories by 'group' it is meant a group object for which all three of $*$, $(\_)^{-1}$, and $e$ are *deterministic*. > [!basicproperties] The category $\mathsf{Grp}$. > 1. Trivial groups $(e)$ are both [[terminal object|initial]] and [[terminal object|final]] objects in $\mathsf{Grp}$. >2. The [[direct product of groups]] $G \times H$ (formed as the [[cartesian product]] endowed with component-wise multiplication) is, in fact, a [[categorical product]] in $\mathsf{Grp}$. >3. The [[free product of groups]] $G * H$ turns out to be the [[categorical coproduct|coproduct]] in $\mathsf{Grp}$. ^properties > [!proof] Proof of Properties. > > 1. To see that trivial groups $(e)$ are initial, recall that [[group homomorphisms take identities to identities and inverses to inverses]], meaning that if $G$ is a [[group]] the only [[group homomorphism|homomorphism]] from $(e)$ to $G$ is $e \mapsto e_{G}$. And the trivial groups $(e)$ are final because the only [[group homomorphism]] from a [[group]] $G$ to $(e)$ is obviously the [[group homomorphism|trivial homomorphism]]. > 2. See [[direct product of groups]]. > 3. See [[free products are coproducts in Grp]] > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```