Questions tagged [slice-category]

For questions regarding slice categories.

Filter by
Sorted by
Tagged with
0 votes
1 answer
47 views

When performing the "opposite" operation in a category, can you reverse arrows that are part of an object's definition?

I'm reading Emily Riehl's Category Theory in Context and pondering Exercise 1.2.i, where the author writes "Defining $\mathcal{C}/c$ to be $(c/\mathcal{C}^{op})^{op}$," in an effort to ...
RicLouRiv's user avatar
  • 107
1 vote
1 answer
46 views

Category of monoids isomorphic to coslice category

Let $k$ be a commutative ring and $R$ a $k$-algebra. The category $R\text{-Bimod}$ of $R$-bimodules becomes a monoidal category with the tensor product of $R$-bimodules. Denote the category of $k$-...
Margaret's user avatar
  • 1,777
2 votes
0 answers
79 views

Left and right adjoints of functor category inclusion

Question. Let $i:\mathcal{C}\hookrightarrow\mathcal{D}$ be a full subcategory. Assume we are given a cocomplete category $\mathcal{A}$. Show that the induced pre-composition functor $i^*:\mathrm{Fun}(\...
Robert's user avatar
  • 528
2 votes
1 answer
109 views

Does The Dependent Product Functor Relate To Type Theory?

I'm confused by the naming of the dependent product functor. As far as I understand, in the category of sets, given $f:X\to Y$ and $u:A\to X$, the dependent product functor $\Pi_f:\mathsf{Set}_{/X}\to\...
fweth's user avatar
  • 3,574
1 vote
1 answer
44 views

Is an epimorphism fibered in contractible kan complexes an acyclic kan fibration?

Suppose $f : X \to Y$ is a morphism of simplicial sets which is degreewise surjective and where the fiber over any vertex of $Y$ is an acyclic kan complex. Is $f$ necessarily an acyclic kan fibration? ...
Brendan Murphy's user avatar
1 vote
1 answer
147 views

How could stalk functor be seen as the pullback functor for the morphism $i_x:\{x\}\to X$?

In the stacks project website, the proof of lemma 6.27.3, https://stacks.math.columbia.edu/tag/0099 , it claims that the stalk functor $[-]_x:\mathscr{F}\mapsto \mathscr{F}_x$ could be seen as the ...
Frank's user avatar
  • 141
1 vote
0 answers
60 views

Representable presheaves on the slice category

$\def\sfC{\mathsf{C}} \def\op{\mathrm{op}} \def\set{\mathsf{Set}} \def\psh{\operatorname{PSh}} \def\ob{\operatorname{Ob}} \def\hom{\operatorname{Hom}}$Let $\sfC$ be a category. A presheaf over $\sfC$ ...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
62 views

Categories fibered in groupoids and the slice category

I read that one example of categories fibered in groupoids is the slice category $\mathcal{C}_{/x}\to \mathcal{C}$ where $x\in \mathcal{C}$ an object. But as the usual definition of slice category ...
Chanel Rose's user avatar
0 votes
1 answer
187 views

Slice model category and initial/terminal objects

Consider $(M,Fib,Cof,WE)$ a model category, that is a complete/cocomplete category with 3 wide subcategories satisfying certain properties. Now consider $X\in\operatorname{Ob}M$ and the slice category ...
raisinsec's user avatar
  • 439
2 votes
3 answers
102 views

What's the name of an object in the slice category?

What do you call an object of the slice category? I have been calling them "slices" but this seems to be wrong: it seems that people use the word "slice" as a synonym of "...
dremodaris's user avatar
6 votes
2 answers
217 views

What exactly are the arrows in a slice category?

I try to wrap my head around the construction of the slice category over an object. I keep reading that an arrow in a slice category (say $\mathbf{D}/D$) between objects $ h \colon X \rightarrow D $ ...
puck29's user avatar
  • 438
1 vote
1 answer
156 views

Adjoints of projections from comma categories

Let $1$ be the terminal category. Consider functors $1 \overset{C}{\rightarrow} \mathcal{C} \overset{F}{\leftarrow} \mathcal{D}.$ I would like to know if there are simple conditions (like presence/...
Richard Southwell's user avatar
0 votes
1 answer
87 views

Meaning of being isomorphic over/under $c$ in category theory

To the point: What does being isomorphic over/under an object $c$ mean? Let $x,y,c$ be objects in a category $\mathsf C$, and let's concentrate on "over". I think "$x\cong y$ over $c$&...
Milten's user avatar
  • 7,031
1 vote
1 answer
656 views

Intuition for comma category and slice category? [closed]

As usual, I am not happy with the definitions until I dont gain some intuition. Could you help me and maybe provide some examples of the concepts? Thank you! My questions are: 1 Comma category = "...
Tereza Tizkova's user avatar
8 votes
1 answer
207 views

Can we have a category of compactifications?

Is there any work on compactifications of spaces in terms of category theory? I would like to know whether there is a defined category of compactifications; I will denote it Comp. Could you describe ...
Tereza Tizkova's user avatar
2 votes
0 answers
58 views

Problem with Awodey's Category Theory Exercise 8.11, showing Sets/X is cartesian closed using a specific hint.

Show that every slice category $\mathbb{Sets}/X$ is cartesian closed. Calculate the exponential of two objects $A \to X$ and $B \to X$ by first determining the Yoneda embedding $y : X \to \mathbb{Sets}...
AprilGrimoire's user avatar
0 votes
1 answer
91 views

(non-)Existence of an endofunctor of $F$ on $g/{\bf B}G^\circlearrowleft$ given the definition of $F$ on objects.

In few words: given an element of a group $g\in G$ given an I'd like to define and endofunctor $$F:g/{\bf B}G^\circlearrowleft\to g/{\bf B}G^\circlearrowleft$$ I have defined $F$ on objects and I'd ...
MphLee's user avatar
  • 2,472
3 votes
0 answers
52 views

base change functor preserves internally projective objects

Let $E$ be a topos . an internally projective object is an object $P$ for which $(-)^P:E\rightarrow E$ preserves epi morphisms. Equavalently: $E/P\rightarrow E$ preserves epis. Prove that for ...
user850424's user avatar
1 vote
0 answers
82 views

Is there a free locally monoidal category?

One of the simpler free monoids is the list. I think a list category would have lists of objects as objects and its maps would also be lists. A list of maps ...
Molly Stewart-Gallus's user avatar
1 vote
0 answers
291 views

Right adjoint of pullback functor

Let $f:A {\rightarrow} B$ be an arrow in a topos $\mathbb{C}.$ The pullback functor $f^*: \mathbb{C} /B {\rightarrow} \mathbb{C} /A$ sends an object of the slice category $\mathbb{C} /B$ to the ...
Richard Southwell's user avatar
4 votes
2 answers
300 views

When is a category the colimit of its slice categories

My question: is it true in general for a small category $\mathcal{C}$ that the canonical functor \begin{aligned} \operatorname{colim}_{c\in \mathcal{C}}\mathcal{C}/x \to \mathcal{C} \end{aligned} is ...
Bastiaan Cnossen's user avatar
2 votes
1 answer
117 views

What is the meaning of a monomorphism in $S/X$ being "fiberwise"?

Notation: $S$:the category of abstract sets; $S/X$: the slice category of $S$ over a set $X$; $A_x$: the fiber of a set $A$ over an element $x$ of the codomain of a function $A\rightarrow X$. My ...
Jaspreet's user avatar
  • 759
1 vote
1 answer
65 views

Are morphisms in a slice category $c/C$ inherited from the original category $C$?

Going through Emily Riehl's Category Theory in Context and something keeps tripping me up. The notation for slice categories, which reminds me of factor group notation in group theory, indicates to ...
Alena Gusakov's user avatar
3 votes
1 answer
155 views

Why is el(-)=$\int(-)$ a functor from functors to a slice cateory?

I am reading up a little on category theory. I am trying to solve Riehl's problem (Category Theory in Context) 2.4.vii on page 72. I believe that it should be quite easy, but it appears to me that ...
Caroline's user avatar
  • 407
11 votes
1 answer
452 views

What is exponentiation in the slice category?

I'm looking for an explicit construction, however involved, of the exponential objects in the slice categories of a topos. If we have a (elementary) topos $\mathbf{C}$ (cartesian closed+subobject ...
augustoperez's user avatar
  • 3,216
2 votes
1 answer
1k views

Trying to understand the definition of a universal property

Here's the definition of a universal property in Wikipedia: (where $U:D\to C$ is a functor and $X$ is an object in $C$) A terminal morphism from $U$ to $X$ is a final object in the category $(...
Abhimanyu Pallavi Sudhir's user avatar
1 vote
1 answer
315 views

Slice category and free objects

*The slice category or over category $C/c$ of a category $ C $ over an object $ c∈C $ has objects that are all arrows $ f∈C $ such that $ cod(f)=c $, and morphisms $ g:X→X'∈C $ from $ f:X→c $ to $...
Imane's user avatar
  • 31
2 votes
0 answers
52 views

In a category ${\rm FinSet}/S$, what operation could find the cospan with smallest domain at the coapex?

Suppose I am working in the ${\rm FinSet}$ category, and suppose a finite set $S$, say, containing a finite amount of character strings. Consider now the slice category ${\rm C} = {\rm FinSet}/S$. ...
Laurent LA RIZZA's user avatar
1 vote
1 answer
206 views

How does the forgetful functor from $\mathbf{C}/C$ to $\mathbf{C}$ forgets the object $C$?

First, sorry for duplication. I've noticed How does the functor $F: \textbf{C}/C \to \textbf{C}$ "forget about the base object" $C$?, but answers there didn't solve my confusion, and my ...
Kinono's user avatar
  • 13
3 votes
0 answers
129 views

Cartesian closed coslice categories

General question: When is a coslice of a cartesian closed category cartesian closed? While slices of cartesian closed categories are often themselves cartesian closed, coslices are rarely ...
Geoffrey Trang's user avatar
2 votes
1 answer
151 views

Filtered colimit and directed colimit, Example

This is a theorem in Locally presentable and accessible categories, by Adamek, Rosicky. For every (small) filtered category $D$ there exists a (small) directed poset $D_0$ and a cofinal functor $H:...
Bryan Shih's user avatar
  • 9,518
1 vote
2 answers
54 views

What does a morphism in the category $\mathsf{C}_{\alpha, \beta}$ correspond to?

Start with a given category $\mathsf{C}$ and fix two morphisms $\alpha: A \to C$ and $\beta: B \to C$. Consider the category $\mathsf{C}_{\alpha, \beta}$ as follows: Does a morphism in this ...
user5826's user avatar
  • 12k
2 votes
2 answers
1k views

Completeness of over category (slice)

Let $\mathcal J$ be a (small) category (denote $I:= \mathcal J_0$) and $\mathcal C$ a category that has all (small) limits (all limits of shape $\mathcal J$ for all $\mathcal J$). Prop 3.4 states then ...
AlvinL's user avatar
  • 8,664
1 vote
1 answer
287 views

How to find initial object of the category of pointed rings?

I have the category of pointed rings. Objects are all pairs $(R, r)$ where $R$ - ring (with 1) and r is the element of R. Morphisms are homomorphism of rings. Morphism $(R, r) \longrightarrow (R', r')$...
Ivan Sharapenkov's user avatar
2 votes
1 answer
120 views

Is $\mathbf{Cat}/\mathcal{C}\simeq\mathbf{Cat}^{\mathcal C^\mathrm{op}}$ when $\mathcal C$ is a $1$-category?

Given a small set $S$, we can define the overcategory $\mathbf{Set}/S$ to be the category whose objects are pairs $(A:\mathbf{Set},a:A\to S)$ and whose morphisms $(A,a)\to(B,b)$ are functions $f:A\to ...
Oscar Cunningham's user avatar
0 votes
0 answers
158 views

How is this a commutative diagram?

So, I'm rather new to category theory (well, Abstract Algebra in general as well), so I decided to pick up Paulo Aluffi's "Algebra: Chapter 0". However, there is one example in the introductory ...
Michael's user avatar
  • 2,609
5 votes
1 answer
753 views

What is the product in a slice category like?

I understand that the product in a slice category $\mathscr{C}/X$ is the pullback in the category $\mathscr{C}$ - my end goal is to prove that fact. The one thing that is (currently) making me bash my ...
PrincessEev's user avatar
  • 43.9k
0 votes
1 answer
341 views

Proving that a slice category is still a category

I recently came across the concept of a category, and I'm trying to make order of some stuff in my mind. Given an object $c$ of a category $\mathbf C$, we define the slice category $c/\mathbf C$ to ...
user avatar
3 votes
1 answer
177 views

Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$

I'm self studying from Category Theory in Context by Emily Riehl and have encountered the following question regarding slice categories: Let $\text{C}$ be a category and $c$ be an object in $\text{...
kelly maggs's user avatar
2 votes
2 answers
353 views

An elementary question about slice categories

I'm just beginning to dabble into some category theory (from Aluffi's Algebra) and I have some difficulty with morphisms in a slice category. Particularly, I can't see why the diagram of a morphism ...
nek28's user avatar
  • 153
4 votes
1 answer
85 views

$\Pi_f$ for a morphism $f$ between simplicial sets

This nLab article says presheaf categories (including $\mathsf{sSet}$, the category of simplicial sets) are locally cartesian closed. For presheaf categories, it can be proven by use of the category ...
aaa's user avatar
  • 443
2 votes
0 answers
186 views

fibered category definition

Im reading Vistolis "Notes on Grothendieck topologies, fibered categories and descent theory", and when i read the definition of a fibered category, it struck me as very similiar to the definition of ...
SuddenlyNotHorrific's user avatar
2 votes
1 answer
117 views

Is the overcategory $C_{/p}$ a subcategory of the $\infty$-category $\operatorname{Fun}(K^{\vartriangleleft}, C)$?

Let $p: K \to C$ be a simplicial morphism from a simplicial set $K$ to an $\infty$-category $C$. $\newcommand{\catSSet}{\mathtt{SSet}}\DeclareMathOperator{\Fun}{Fun}$ The over $\infty$-category $C_{/p}...
Fang Hung-chien's user avatar
1 vote
1 answer
312 views

Dual of a Comma Category [duplicate]

Let $F:\mathcal{C} \longrightarrow \mathcal{D}$ be a functor between two categories and $d$ be an object of $\mathcal{D}$. Does the following hold: $(F \downarrow d)^{op} \cong (d \downarrow F^{op})$...
Balletti's user avatar
  • 119
2 votes
1 answer
168 views

Comma categories of locally finitely presentable categories

Let $\mathbf{C}$ be a locally finitely presentable category, and let $A$ be an object of $\mathbf{C}$. The slice category $\mathbf{C}/A$ is locally finitely presentable. Is this also true for the co-...
JJ1993's user avatar
  • 406
1 vote
0 answers
142 views

Slice category - origin of the name

Where does the name "slice category" originate for the category $(\mathscr{C} \downarrow C)$ of objects over $C$?
user50229's user avatar
  • 3,082
1 vote
3 answers
329 views

How does the functor $F: \textbf{C}/C \to \textbf{C}$ "forget about the base object" $C$?

Let $\textbf{C}/C$ be a slice category with base object $C$. The functor that comes to mind is the one such that objects $f:X \to C$ are mapped to $X$, and arrows $a: X \to X'$ are mapped to ...
Daniel Donnelly's user avatar
8 votes
1 answer
434 views

Slice and comma categories in French

I have to give a presentation in French on category theory, and most of the literature I use is in English. This is not too much of a problem since most of the terms translate fairly easily. However, ...
Arnaud D.'s user avatar
  • 20.9k