Primo esercizio(banale) su Categorie

da **anto_zoolander** » 31/07/2019, 20:48

Ciao!

So che è un esercizio banale però vorrei cominciare a prendere dimestichezza con lo strumentario e le terminologie

Sia $ \mathrm{C} $ una categoria
Se $ 0,0' \in \mathrm{Obj_{C}} $ sono due oggetti iniziali allora sono isomorfi

Poiché sono oggetti iniziali allora esistono e sono unici i morfismi

$ \mathrm{ i \in Hom_{C}(0,0’), j \in Hom_{C}(0’,0)} $

Data la legge si composizione $ \mathrm{ i \circ j \in Hom_{C}(0’,0’), j \circ i \in Hom_{C}(0,0)} $

Dato che sono sempre oggetti iniziali queste due ultime classi contengono soltanto le identità da cui per l’unicità

$ \mathrm{ i\circ j= id_{0’} \wedge j\circ i=id_{0} } $

Quindi entrambi i morfismi sono isomorfismi.

Un esempio che può venirmi in mente è $ \mathrm{Set} $ dove un elemento iniziale è $emptyset$

da **marco2132k** » 31/07/2019, 22:06

Ciao. Esiste un'unica freccia $ 0\to o $ per ogni $ o\in\operatorname{Ob}C $, e per ogni oggetto $ o $ deve esistere almeno $ \operatorname{id}_o $; quindi è giusto.

Puoi vedere la definizione di oggetto terminale (o iniziale, che è lo stesso) contestualizzata qui, ad esempio. (Ma sono convinto che ci siano altri mille mila usi).

da **anto_zoolander** » 31/07/2019, 22:16

Perfetto.
Diciamo che non è stato così pessimo l’approccio alle categorie, anche se ancora sono spaesatuccio :lol:

Ti ringrazio per la referenza gli ho dato un’occhiata e mi piace molto; appena torno me la leggo per bene.
Grazie ancora!

da **Indrjo Dedej** » 01/08/2019, 15:15

Ah, non ti preoccupare, la strada è in salita...

Il link di marco è utile quando si parla di teoria delle categorie come rimpiazzo, o meglio come nuova, teoria degli insiemi. Per me, se lo leggi adesso, fai bene a rileggerlo a distanza di tempo. Dovresti conoscere qualche assiomatica per la teoria degli insiemi, per poter capire le ragioni di alcune scelte.
Il mio consiglio per iniziare è il Leinster (pdf, ma su Amazon trovi sicuramente il cartaceo).
Però se i tuoi interessi sono fondazionali potresti considerare Topoi di Robert Goldblatt, il quale parla distesamente della teoria delle categorie, per poi indirizzarla in un certo modo.
Poi c'è il canale Youtube TheCatsters.

Tornando all'esercizio banale, rifletti/riflettete (anche in relazione a questo) su "esiste uno e un solo" e "esiste un unico". Attenzione...

da **Indrjo Dedej** » 01/08/2019, 15:32

marco2132k ha scritto:Ma sono convinto che ci siano altri mille mila usi

Prendi una proprietà universale e vedi se ci sono iniziali o terminali. :wink:

da **caulacau** » 02/08/2019, 12:57

Ogni oggetto universale è un oggetto iniziale o terminale di una opportuna categoria.

Copincollo da un libro che sto finendo.

Definition (Cone completions of $J$).
Let $J$ be a small category; we denote $J^\rhd$ the category obtained adding to $J$ a single terminal object $\infty$; more in detail, $J^\rhd$ has objects $J_o \cup\{\infty\}$, where $\infty\notin J$, and it is defined by
\[\begin{align*}
\hom_{J^\rhd}(j,j') & = \hom_J(j,j') \\
\hom_{J^\rhd}(j,\infty) & = \{*\}
\end{align*}
\] and it is empty otherwise. This category is called the right cone of $J$.[/center]

Dually, we define a category $J^\lhd$, the left cone of $J$, as the category obtained adding to $J$ a single initial object $-\infty$; this means that $\hom_{J^\lhd}(j,j')=\hom_J(j,j')$, $\hom_{J^\lhd}(-\infty, j)=\{*\}$, and it is empty otherwise.

Remark.
The correspondences $J\mapsto J^\rhd$ and $J\mapsto J^\lhd$ are, obviously, functorial. As an easy exercise, define them on morphisms and prove their functoriality. Prove also that there is a natural embedding of $J$ into $J^\rhd$ and one into $J^\lhd$ (that we invariably denote $i$ in the following discussion).

Definition (Cone of a diagram).
Let $J$ be a small category, $\mathcal{C}$ a category, and $D : J \to \mathcal{C}$ a functor; an idiosyncratic way to refer to it will be as a diagram of shape $J$. We call a cone for $D$ any extension of the diagram $D$ to the left cone of $J$, so that the diagram

$\xymatrix{ J \ar[r]^D \ar[d]_{i_J} & \mathcal{C} \\ J^\lhd\ar@{.>}[ur]_{\bar D} }$

commutes.
Every such extension is thus forced to coincide with $D$ on all objects in $J\subseteq J^\lhd$; the value of $\bar D$ on $-\infty$ is called the base of the cone; dually, the value of an extension of $D$ to $J^\rhd$ coincides with $D$ on $J\subseteq J^\rhd$, and $\bar D(\infty)$ is called the tip of the cocone.

Cones for $D$ are exactly cocones for the opposite functor $D^\text{op}$.

Proposition.
The class of cones for $D$ forms a category $\mathsf{Cn}(D)$, whose morphisms are the natural transformations $\alpha : D'\Rightarrow D'' : J \to \mathcal{C}$ such that the right whiskering of $\alpha$ with $i : J \to J^\rhd$ coincides with the identity natural transformation of $D$.

Definition (Colimit, limit).
The limit of a diagram $D : J\to \mathcal{C}$ is the terminal object ''$\lim_J D$'' in the category of cones for $D$; dually, the colimit of $D$ is the initial object ''$\text{colim }_J D$'' in the category of cocones for $D$.
Unwinding the above definition we get the more classical definition of a limit and of a colimit:

A cone for a diagram $D : J \to \mathcal{C}$ is a natural transformation from a constant functor $\Delta_c : J \to \mathcal{C}$ to $D$;

there is a category of cones for $D$, where morphisms between a cone $c\to D$ and a cone $c'\to D$ are arrows $k : c\to c'$ such that the diagram

$\xymatrix{ Di \ar[dd]_{D\phi}&& \\ &c\ar@{.>}[r]^k \ar[ul]^{l_i}\ar[dl]_{l_j}& c^\prime \ar[ull]_{l_i^\prime}\ar[dll]^{l_j^\prime}\\ Dj }$

is commutative;

a limit for $D$ is a terminal object in the category of cones for $D$. This means that given a cone for $D$, there is a unique arrow $k$ which is a morphism of cones.

Of course, a straightforward dualization yields the definition of a cocone, and a colimit for $D$.

[...]

In the present subsection we collect some examples of co/limit: the reader is invited to work out the details of the construction of each limit and colimit of the following diagrams in the category of sets and functions first, and then in some familiar categories of algebraic structures (e.g. groups, vector spaces...).

Example (Product).
Let $J$ be a set, and $\{X_i \mid i\in J\}$ a family of objects of a category $\mathcal{C}$; the product of the $\{X_i\}$, denoted $\prod_{i\in J} X_i$, is the limit of the diagram $D : J \to \mathcal{C}$, when the set $J$ is regarded as a discrete category.

The universal property exhibited by the object $\prod_{i\in J} X_i$ is the following: there is a cone $\{p_i : \prod X_j \to X_i\mid i\in J\}$ such that for every other cone $\{f_i : E\to X_i \mid i\in J\}$ there exists a unique dotted $\bar f : E \to \prod_{i\in J} X_i$ such that

$\xymatrix{ E \ar@{.>}[d]_{\bar f}\ar[dr]^{f_{\bar\imath}} & \\ \prod X_i \ar[r]_{p_{\bar\imath}}& X_{\bar\imath} }$

commutes for every $\bar\imath \in J$.

Example (Pullback).
Let $J$ be the category $0\to 2\leftarrow 1$, and $\{X_0 \to X_2 \leftarrow X_1\}$ the corresponding diagram $X : J\to \mathcal{C}$; the pullback of the diagram $X$, denoted $X_0 \times_{X_2} X_1$, is the limit of $X$; the universal property exhibited by the object $X_0 \times_{X_2} X_1$ is the following: there is a cone $X_0 \xleftarrow{p_0} X_0\times_{X_2}X_1 \xrightarrow{p_1} X_1$ suc that for every other cone $X_0 \xleftarrow{f_0} E \xrightarrow{f_1} X_1$ there exists a unique dotted $\langle f_0,f_1\rangle$ such that

$\xymatrix{ E \ar@{.>}[dr]\ar@/^1pc/[drr]^{f_0}\ar@/_1pc/[ddr]_{f_1}&& \\ & X_0 \times_{X_2} X_1 \ar[d]_{p_1} \ar[r]^{p_0} & X_0\ar[d]^{q_{02}} \\ & X_1 \ar[r]_{q_{12}}& X_2 }$

In the same notation above, when we want to stress the dependence of the pullback from the maps $q_{02}, q_{12}$, the object is sometimes denoted as $q_{02}\times q_{12}$ instead of $X_0 \times_{X_2} X_1$.

Example (Equalizer).
Let $J$ be the category $0\rightrightarrows 1$, and $\{X_0 \underset{v}{\overset{u}\rightrightarrows} X_1\}$ the corresponding diagram $X : J\to \mathcal{C}$; the equalizer of the diagram $X$, denoted $\text{eq}(u,v)$, is the limit of $X$; the universal property exhibited by the object is the following: there is an cone $ e : \text{eq}(u,v) \to X_0$ and for every other cone $k : E \to X_0$ there is a unique dotted $\bar k : E \to \text{eq}(u,v)$ such that

$\xymatrix{ \text{eq}(u,v) \ar[r]^e & X_0 \ar@<3pt>[r]^u \ar@<-3pt>[r]_v & X_1 \\ E\ar[ur]^k \ar@{.>}[u]_{\bar k} }$

Example (Examples of colimits)
Each of the above examples can be dualized: in fact, a colimit in $\mathcal{C}$ is no more, no less than a limit in $\mathcal{C}^\text{op}$, and viceversa, a limit in $\mathcal{C}$ is no more, no less than a colimit in $\mathcal{C}^\text{op}$.

Let $J$ be a set, and $\{X_i \mid i\in J\}$ a family of objects of a category $\mathcal{C}$; the coproduct of the $\{X_i\}$, denoted $\coprod_{i\in J} X_i$, is the colimit of the diagram $D : J \to \mathcal{C}$, when the set $J$ is regarded as a discrete category.

The universal property exhibited by the object $\coprod_{i\in J} X_i$ is the following: there is a cocone $\{q_i : X_i \to \coprod X_j \mid i\in J\}$ such that for every other cocone $\{f_i : X_i \to E \mid i\in J\}$ there exists a unique dotted $\bar f : \coprod_{i\in J} X_i \to E$ such that

$\xymatrix{ E \ar@{<.}[d]_{\bar f}\ar@{<-}[dr]^{f_{\bar\imath}} & \\ \coprod X_i \ar@{<-}[r]_{q_{\bar\imath}}& X_{\bar\imath} }$

commutes for every $\bar\imath \in J$.

Let $J$ be the category $1\to 0\leftarrow 2$, and $\{X_1 \to X_0 \leftarrow X_2\}$ the corresponding diagram $X : J\to \mathcal{C}$; the pushout of the diagram $X$, denoted $X_1 \amalg_{X_0} X_2$, is the colimit of $X$; the universal property exhibited by the object $X_1 \amalg_{X_0} X_2$ is the following: there is a cocone $X_1 \xrightarrow{q_1} X_1\amalg_{X_0}X_2 \xleftarrow{q_2} X_1$ such that for every other cocone $X_1 \xrightarrow{f_1} E \xleftarrow{f_2} X_2$ there exists a unique dotted $ [f_0,f_1]$ such that

$\xymatrix{ E \ar@{<.}[dr]\ar@{<-}@/^1pc/[drr]^{f_0}\ar@{<-}@/_1pc/[ddr]_{f_1}&& \\ & X_0 \amalg_{X_2} X_1 \ar@{<-}[d]_{p_1} \ar@{<-}[r]^{p_0} & X_2\ar@{<-}[d] \\ & X_1 \ar@{<-}[r]& X_0 }$

Define coequalizers as an exercise.