Tagged Questions

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How would you describe category $\mathsf{Rel}$?

I encountered two definitions for a category denoted by $\mathsf{Rel}$: Objects are pairs $\left(A,R\right)$ where $A$ is a set and $R$ a relation on $A$. Arrows in ...
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Notation for a functor between comma categories

Suppose we have two categories $D$ and $S$, as well as two functors $K,L:D\to S$ and a natural transformation $\varphi:K\to L$. Given another category $C$ and a functor $Y:C\to S^D$, is there a nice ...
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What is $1 / \mathbf{Set}$ if $1$ is a one-element set and $\mathbf{Set}$ a category?

What does $1 / \mathbf{Set}$ denote? A pointed set is a set $X$ equipped with an element (a basepoint) $x \in X$. Let $\mathbf{Set_*}$ be the category of pointed sets and basepoint-preserving ...
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Standard Notation For The Set of All the Morphisms Of A Category

Let $\mathscr C$ be a category. Let $\text{Ob}(\mathscr C)$ be the set of all the objects of $\mathscr C$. Is there a standard notation for $\bigcup_{A,B\in\text{Ob}(\mathscr C)}\text{Mor}(A,B)$? ...
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Notation for the subcategory of commutative $R$-algebras

Let $R$ be a commutative ring (with identity) and let $R\mathbf{Alg}$ denote the category of $R$-algebras. My question: Is there a suitable notation for the full subcategory of commutative ...
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Notation for the set of monomorphisms in $\mathrm{Hom}(A,B)$

Let $\mathcal{C}$ be a small category, and let $A$ and $B$ be objects in $\mathcal{C}$. Is there any standard notation for the subset of all monomorphisms $A\hookrightarrow B$ in $\mathrm{Hom}(A,B)$? ...
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function application order

In traditional mathematics, when we post-compose $x$ by $f$ we write $fx$ or $f(x)$, that is we prefix writing things right to left. I realize some might be used to it, and it is absolutely trivial, ...
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Why an integral symbol for the category of elements of a presheaf?

Let $\mathbf C$ be a category and $P \colon \mathbf C^{\rm op} \to \mathbf{Set}$ a presheaf. One can associate to $P$ the category of elements of $P$ (also called Grothendieck construction over $P$), ...
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A question about co-exponentials

An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right): \mathcal{C} \rightarrow Set$$ or equivalently, as the terminal object of ...
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Where can I learn more about these subcategories of functor categories?

Note: I've substantially edited the definition; $f$ is now allowed to be a functor. Given categories $\mathcal{C}$ and $\mathcal{D}$, we can form the functor category $[\mathcal{C},\mathcal{D}]$. Now ...
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Nice notation for projection maps

Let $X\times Y$ be a product of two object of a category, and consider the natural projections $$X\times Y \to X \quad\text{ and }\quad X\times Y \to Y.$$ Usually I denote them by $\pi_X$ and ...
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Name for a category

Is there any name or notation for this category? Let $U$ be a set. By "function" I will mean a function $U\rightarrow U$. objects are functions; morphisms from a function $A$ to a function $B$ are ...
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Notation for “parallel” morphisms in a diagram

Suppose $f\colon A\to B$ and $g\colon A\to B$ are possibly-distinct morphisms. How do I stick them both in a diagram (along with, e.g., their (co)equalizer) without suggesting that they are equal?
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Category of topological pairs

Is there a standard abbreviation for the category of topological pairs? I have searched for it in vain.
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A definition check.

In Daniele Turi's "Category Theory Lecture Notes" from the University of Edinburgh, shouldn't "cone" be "cocone" in the definition of a colimit? A generic arrow $\tau : J\Rightarrow \Delta Y$ from ...
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ENS is an abbreviation of?… [duplicate]

In CWM Mac Lane uses the term $\mathbf {ENS}$ for a category having as objects the subsets of a given set and as morphisms the functions from these sets to these sets. What is abbreviated by the ...
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What is principal ideal and why is it isomorphic to a slice category?

Stuck again. Not even a page through :-( Reading Awodey's Category Theory [p.17] he says (this is after the definition of what a slice category $\boldsymbol{C}/C$ is): If $C=P$ is a poset ...
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How does slice category help create functor?

Reading Awodey [p.16-17], he states the following: The slice category $\boldsymbol{C}/C$ of a cateogry $\boldsymbol{C}$ over an object $C\in\boldsymbol{C}$ has [definition of slice category ...
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Category formulas without explicit specifying of objects

Consider the following example: $C$ is a category each of whose Hom-sets is partially ordered. Let $f$, $g$, and $h$ are morphisms of this category. Consider the formula: $g\circ f \ge h$. Intuition ...
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Functors preserving (commuting with) exponentials

I have been unable to find any established names for functors preserving exponential objects in general ($F$ such that $F(A^B) \cong FA^{FB}$) and/or those "commuting" with functors $-^A$ (some ...
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Is there a meaningful distinction between “inclusion” and “monomorphism”?

The title pretty much says it all. As far as I can tell, the terms "inclusion" and "monomorphism" are equivalent. (Ditto for $\hookrightarrow$ and $\rightarrowtail$.) Is this the case? Edit: ...
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What does $\Rightarrow$ in the definition of a natural transformation mean?

In the definition of a natural transformation from http://ncatlab.org/nlab/show/natural+transformation it is said that a natural transformation is $\alpha : F \Rightarrow G$ (and the definition ...
I've stumbled upon the definition of exact sequence, particularly on Wikipedia, and noted the use of $\hookrightarrow$ to denote a monomorphism and $\twoheadrightarrow$ for epimorphisms. I was ...