Questions tagged [slice-category]

For questions regarding slice categories.

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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 ...
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83 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 ...
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1answer
56 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 ...
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1answer
37 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 ...
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1answer
56 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 ...
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1answer
152 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 ...
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1answer
263 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 $(...
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1answer
103 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 $...
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38 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 ${\rm C} = {\rm FinSet}/S$. ...
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1answer
87 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 ...
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45 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 ...
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1answer
54 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:...
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2answers
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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 ...
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1answer
219 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 ...
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1answer
146 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 $(R, r) \longrightarrow (R', r')$...
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1answer
89 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 ...
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88 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 ...
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1answer
242 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 ...
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1answer
101 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 ...
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1answer
97 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{...
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2answers
237 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 ...
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1answer
69 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 ...
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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 ...
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1answer
108 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}...
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170 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: $(F \downarrow d)^{op} \cong (d \downarrow F^{op})$...
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1answer
91 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-...
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104 views

Slice category - origin of the name

Where does the name "slice category" originate for the category $(\mathscr{C} \downarrow C)$ of objects over $C$?
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3answers
292 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 ...
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297 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, ...