Questions tagged [terminal-objects]
14 questions
12
votes
2 answers
Do Wikipedia, nLab and several books give a wrong definition of categorical limits?
It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and write some errata emails. A standard definition of a…
joriki
- 243,700
7
votes
2 answers
Is the terminal object always cofibrant?
Let $\mathcal{M}$ be a model category. Is the unique map from the initial to the terminal object $!: \mathbf{0} \rightarrow \mathbf{1}$ always a cofibration? If it helps, we can even assume that $\mathcal{M}$ is combinatorial (i.e. locally…
chickenNinja123
- 849
5
votes
0 answers
Does the (2,1)-category of graphs have a terminal object
I am considering the $(2,1)$ -category of Graphs as defined by Tien Chich and Laura Scull in "A homotopy category of graphs". Its objects are single-edged non-directed graphs, and the $1$ -morphisms are the usual graph morphisms. In this $1$…
Virge
- 155
4
votes
1 answer
Is there the terminal object in the category of integral domains?
I wondered what the initial object and the terminal object are in the category of integral domains. Simple argument: Since integral domains do not put an additional restriction on the definition of a ring homomorphism, integral domains should…
Dannyu NDos
- 2,189
3
votes
2 answers
Why the dual of the naturals, a terminal arrow among all $X\to X+1$, is the extended natural numbers object?
When reading about corecursion for the first time (in my case, on nLab , perhaps unwisely), one is confronted with the extended natural numbers example and a blank statement that it is terminal among all $X\to X+1$ arrows. The nLab article on…
Al.G.
- 1,830
2
votes
0 answers
Initial object in category of morphisms $A → Z$
In Algebra Chapter $0$ , for a fixed set $A$ , when talking about "The quotient $A/{\sim}$ is universal with respect to the property of mapping $A$ to a set in such a way that equivalent elements have the same image", he explains that the morphism…
squirrels
- 217
1
vote
1 answer
If a functor $G : C \rightarrow D$ preserves finite limits, then $\operatorname{colim} G$ is a terminal object.
$\newcommand{\colim}{\operatorname{colim}} \newcommand{\Psh}{\operatorname{Psh}} \newcommand{\Elt}{\operatorname{Elt}} \newcommand{\Func}{\operatorname{Func}}$ Let $C$ be a small category with finite limits, and let $D$ be a category with (small)…
Sambo
- 7,714
0
votes
1 answer
How can a constant functor pick out a terminal element from an empty index category?
I have come across the assertion that a terminal object $T$ in a category $\mathcal{C}$ can be understood as the limit of an empty diagram, i.e. a diagram $D:\mathcal{J}\to\mathcal{C}$ whose index category $\mathcal{J}$ is the empty category.…
kjo
- 14,954
0
votes
0 answers
Category Glued to its Dual Category
Suppose I have a category $C$ . Then I can construct its dual $C^\bot$ , by way of a contravariant equvalence $C \to C^\bot$ . Suppose that $C$ has an initial object $0 : C$ so that $0^\bot$ is terminal in $C^\bot$ . What is it called if I construct…
Enid
- 11
0
votes
1 answer
Does the composition of a monic morphism with a terminal morphism make a monic morphism?
Consider $T$ to be a terminal element, $T \stackrel{f}{\rightarrowtail} A$ be a monic morphism (this can be shown by the terminal property of $T$ ) and $B\stackrel{\tau_B}{\rightarrow} T$ be the unique terminal morphism from $B$ to $T$ . Then does…
Felipe Dilho
- 187
0
votes
1 answer
How do I prove the unicity of the terminal morphism obtained by being a equivalent monomorphism to one which has a terminal object as domain?
To clarify, consider in a category $\mathbf{C}$ , a object $B$ and a terminal object $T$ both in Obj $_\mathbf{C}$ , and a monomorphism $T \stackrel{f}{\rightarrowtail} B$ . If $g$ in Mor $_\mathbf{C}$ is any other monomorphism with $B$ as codomain,…
Felipe Dilho
- 187
0
votes
0 answers
Characterisation of terminal category in the 2-category sense
On nlab https://ncatlab.org/nlab/show/terminal+category , it is stated that a category is terminal in the 2-category sense precisely when it is inhabited and indiscrete. I wanted to try to prove this for myself, and I have got most of the way…
Harry Partridge
- 697
0
votes
1 answer
Presheaf is terminal iff it maps every object to a singleton
I've read the statement in the title in a text as a side note without proof. One way is easy: using Yoneda, terminality of a presheaf $X \in [\mathcal{C}^\text{op}, \textbf{Set}]$ implies that for any other presheaf $F$ , so in particular for any…
Jos van Nieuwman
- 2,152
0
votes
0 answers
In categories with zero objects: completeness $\Leftrightarrow$ cocompleteness?
Suppose we have a small category $\mathcal{I}$ , a diagram $D : \mathcal{I} \to \mathcal{C}$ , and a functor $F : \mathcal{C} \to \mathcal{D}$ . De know that the functor $\hat{F} : \text{Cone}_{\mathcal{C}}(D) \to \text{Cone}_\mathcal{D}(F \circ D)$…
Jos van Nieuwman
- 2,152