Questions tagged [slice-category]

For questions regarding slice categories.

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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 categoy $\mathsf C$, and let's concentrate on "over". I think "$x\cong y$ over $c$&...
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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 = "...
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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}...
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(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 ...
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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 ...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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2 votes
2 answers
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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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$\Pi_f$ 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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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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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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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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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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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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8 votes
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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, ...
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