0
votes
2answers
46 views

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, ...
5
votes
0answers
60 views

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 ...
3
votes
1answer
41 views

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 ...
0
votes
2answers
69 views

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 ...
1
vote
1answer
95 views

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 ...
1
vote
2answers
61 views

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?
3
votes
1answer
99 views

Category of topological pairs

Is there a standard abbreviation for the category of topological pairs? I have searched for it in vain.
1
vote
1answer
71 views

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 ...
0
votes
1answer
60 views

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 ...
0
votes
0answers
39 views

Embedding vs restriction

Embedding is the morphism $( A ; B ; \operatorname{id}_A)$ of the category $\mathbf{Rel}$ for sets $A \subseteq B$. I call restriction the morphism $( A ; B ; \operatorname{id}_B)$ for sets $A ...
2
votes
1answer
47 views

Can objects repeat in commutative diagrams?

Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is ...
2
votes
3answers
52 views

A notation for a morphism in a thin category

Consider a thin category with objects $A\leq B$. There exists a unique morphism $A\rightarrow B$. Is there a standard notation for this morphism (given $A$ and $B$)?
0
votes
2answers
39 views

Is there a convention for precedence of operators in an additive category?

The laws for an additive category are that there must be a zero object, binary products, that every Hom-set is an abelian group, and that the morphism addition distributes over composition. My ...
8
votes
2answers
114 views

Etymology of Tor and Ext

The names of the important functors Tor and Ext seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from.
2
votes
2answers
101 views

useful notation for pullback

Let $f:A\to C\leftarrow B:g$ be morphisms in a category. There exists in literature a useful notation for the morphisms $\bar f:A\times_C B\to B$ and $\bar g:A\times_C B\to A$ in terms of $f$ and $g$? ...
3
votes
2answers
96 views

Relation-preserving maps as morphisms of a category

What is the canonical name for the category whose objects are all pairs $(X, \rho)$, where $X$ is a set and $\rho$ is a binary relation on $X$, and whose objects are relation-preserving maps? That is, ...
4
votes
1answer
140 views

What is the circumstances of switching to the non-mainstream notation for composition of morphisms?

In mainstream notation for composition, by $g \circ f$ we mean $\operatorname{dom} \left(g \circ f \right) = \operatorname{dom}\left(f\right)$ and $\operatorname{cod} \left(g \circ f \right) = ...
1
vote
1answer
96 views

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 ...
4
votes
1answer
145 views

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 ...
1
vote
2answers
60 views

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 ...
0
votes
1answer
75 views

Do function with the following property have special name?

I'm writing "a structure preserving surjection" way too much when I need to refer a function of the following property: $$ Y \subseteq Z, X \subseteq Z. g: Z \to A, g \text{ is some fixed ...
1
vote
1answer
76 views

Graph of a Rel-morphism

Let $F=(f;A;B)$ is a morphism of the category $\mathbf{Rel}$ (the category whose objects are sets and morphisms are defined as binary relations). How to name and how to denote $f$ when we know $F$? ...
1
vote
2answers
93 views

Standardized Notation for Well-Known Categories

It seems that every author has rather personal and unique conventions for designating "well-known" categories. This raises the question: Is there a reference available, on-line or otherwise, that ...
8
votes
1answer
135 views

Exact sequences with parallel arrows

In Milne's √Čtale Cohomology (both the book and the online notes), he sometimes says a diagram of the form $$0 \rightarrow A \rightarrow B \rightrightarrows C$$ is an exact sequence if $A \to B$ is ...
1
vote
1answer
282 views

What are notations to express uniqueness in formulae and diagrams?

I am familiar with the notation $\exists\,!$ to express both existence and uniqueness. For example $$\;\;\exists\,!x\!:\!P\,(x)\;\;$$ means "there exists a unique $x$ such that $P\,(x)$ holds", for ...
2
votes
1answer
87 views

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 ...
4
votes
2answers
267 views

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: ...
0
votes
1answer
293 views

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 ...
8
votes
1answer
386 views

What does **Ens** stand for?

Earlier someone was asking about the category "Ens" described in Categories for the Working Mathematician. My question is more basic: What does Ens stand for? Most of the categories have names that ...
6
votes
3answers
2k views

Special arrows for notation of morphisms

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 ...