3
votes
0answers
45 views

Objects without extensions

How do you call an object $X$ for which every monomorphism $i : X \hookrightarrow Y$ has a retract (i.e.\ a morphism $r : Y \rightarrow X$ such that $r \cdot i = 1_X$)? I think of Y as an extension ...
7
votes
1answer
64 views

Is there a name for those commutative monoids in which the divisibility order is antisymmetric?

Every commutative monoid $M$ is naturally equipped with its divisibility preorder, defined as follows. $$x \mid y \leftrightarrow \exists a(ax=y)$$ Is there a name for those commutative monoids such ...
2
votes
1answer
36 views

Is there a phrase to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?

Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of ...
0
votes
0answers
39 views

A term for category where every loop of morphisms is an identity

"A category where composition of every loop of morphisms is an identity." Moreover, in the case I am thinking about, morphisms are bijective functions. Is there a name for this concept?
2
votes
2answers
56 views

What is meant by a “structure map”?

The title is the question. Somehow I should know the answer, but I am by no means sure what is meant exactly by it. Perhaps it doesn't have a definite meaning and only in context, could someone ...
5
votes
1answer
64 views

Name for a property in a brutally elementary presentation of a monad

For evil reasons of my own, I'm trying to give a presentation of a monad in primitive terms, assuming only the notion of a category. More honestly, I looked at this post and got intrigued by the ...
6
votes
0answers
63 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
1
vote
2answers
60 views

Name for categories in which isomorphic implies equal?

A quick terminology question: Is there any particular name for a category in which each object is uniquely determined by its isomorphism class?
3
votes
1answer
48 views

Rel: the category of relations

$\text{Rel}$ is the standard name for the category of sets and relations. Confusingly in "Abstract and concrete categories" (ACC), page 22, $\text{Rel}$ is defined as a category whose objects are ...
3
votes
1answer
91 views

Is every “almost” isomorphism an isomorphism?

Let $f:A \mapsto B$, $g:B \mapsto A$ and $h:B \mapsto B$ be such that $g \circ f=\operatorname{id}_A$ and $f \circ g \circ h=\operatorname{id}_B=h \circ f \circ g$. Can we conclude ...
3
votes
1answer
105 views

Is a homomorphism expected to be a (structure-preserving) map?

Is a homomorphism a special type of morphism, namely a structure-preserving map? For a morphism (of a category), it is clear that we can't always expect that a morphism is necessarily a ...
3
votes
1answer
46 views

Terminology concerning conjugation in groups of functions.

If there is a function $a$ such that $a\circ g\circ a^{-1}=h$ then the functions $g$ and $h$ are conjugate to each other. If one wished to identify $a$, would one say "$g$ and $h$ are conjugate "by ...
2
votes
0answers
39 views

Concrete categories possessing many forgetful functors

Given a set $X$, is there a name (like $X$-concrete category) for those categories $\mathbf{C}$ equipped with a forgetful functor $F_x : \mathbf{C} \rightarrow \mathbf{Set}$ for each $x \in X$? The ...
0
votes
0answers
28 views

An interleaved path of arrows in a category/digraph

In what I am doing now, a slightly strange concept emerged; My objects are sequences of arrows $\{f_i\}_{i=1}^{n}$ in a small category such that for every $1\leq i\leq n-1$, there is an arrow from the ...
5
votes
1answer
136 views

What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
0
votes
0answers
142 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
1
vote
1answer
53 views

In a dagger category, what do we call an arrow $f$ such that $f \circ f^\dagger \circ f = f$?

In a dagger category, what do we call an arrow $f$ satisfying $f \circ f^\dagger \circ f = f$? In $\mathrm{Rel}$ (and, more generally, an allegory) this is straightforwardly equivalent to $f \circ ...
1
vote
1answer
45 views

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$?

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$? Clearly, every such arrow is a split monomorphism; further, if such an $f$ is ...
2
votes
2answers
85 views

Why are left/right adjoint functors not called up/down?

I am studying category theory and I recently learned about adjoint pairs of functors. It seems to me that they are called left and right adjoints because if we have categories $\mathcal{C}$ and ...
4
votes
1answer
72 views

Have arrows in a category with this property a special name?

Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a ...
2
votes
0answers
70 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
2
votes
1answer
57 views

Is there a special name for ring homomorphisms $f : R \rightarrow S$ with $f^*(C(R)) \subseteq C(S)$?

Edit. For some reason, I called the functor $F$ described below a full functor as opposed to a faithful functor. The problem has now been corrected. For any ring $R$, let $C(R)$ denote the center of ...
1
vote
1answer
101 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 ...
4
votes
1answer
84 views

What's the real name for these things? Categories whose morphisms have “length.”

A fairly obvious "categorification" of metric spaces is as follows. First, let us agree to view $\mathbb{R}_+$ as an ordered Abelian monoid, where by "Abelian monoid" we really mean a category whose ...
3
votes
1answer
73 views

Monoids as categories; does this construction have a name?

We can view a monoid $M$ as a category with a single object. However, there is another way to make $M$ into a category. Take the elements of $M$ as objects, and define $\mathrm{Hom}(x,y)$ to be set of ...
2
votes
1answer
65 views

Is this “set quotient” known?

Let $A,B$ be subsets of a set $X$. Then there is a largest subset $C \subseteq X$ such that $C \cap A \subseteq B$. Explicitly, we have $C = \{x \in X : x \in A \Rightarrow x \in B\} = (X \setminus A) ...
0
votes
0answers
41 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 ...
1
vote
0answers
40 views

Rel instead of Set in a concrete category

Concrete category is a pair $( \mathcal{C}; U)$ where $\mathcal{C}$ is a category and $U$ is a faithful functor $\mathcal{C} \rightarrow \mathbf{Set}$. But how to name a a pair $( \mathcal{C}; U)$ ...
1
vote
0answers
59 views

Functor whose values on morphisms are monomorphisms

Is there a name for a functor whose values on morphisms are monomorphisms?
3
votes
1answer
79 views

Completing a Partially Defined Associative Binary Operation

This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the ...
9
votes
1answer
429 views

On a joke of Yoneda embedding

I have heard a joke like this: The Yoda embedding, contravariant it is. And a joke concerning "How to put an elephant into a refrigerator", a comment from "Category Theorist" says Isn’t this ...
0
votes
1answer
61 views

Canonical direct product (in a category)

In some categories there are more than one (isomorphic) direct products: For example in Set there are $A\times B$ and $B\times A$ products (as well as many others). But only one of these products ...
6
votes
1answer
82 views

name of the unit of adjunction between $-\times C$ and $\cdot^C$

Answers to a earlier question about the categorical interpretation of first-order quantification led me to learn more about adjoints. Now, I understand that a category $\mathscr{C}$ with products has ...
0
votes
0answers
82 views

Morphisms in Bourbaki “Theory of Sets”

In Bourbaki "Theory of Sets" there is notion of "morphisms" and different kind of morphisms such as "initial morphisms". These are defined in terms of order theory. It seems that Bourbaki treatment ...
5
votes
1answer
97 views

Terminology for metric space with “anti-symmetric” distance

I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$): $d(x, x) = 0$ $d(x, y) + d(y, z) \geq ...
3
votes
2answers
82 views

Categories without identities

What's the name of "categories without identities", i.e. of digraphs with just an associative binary operation on its "matching" arrows (disregarding identities)?
8
votes
1answer
158 views

A category where maps are factorizations - what is this called?

Let $\mathcal C$ be a category, and define $\mathcal D$ to be the category whose objects are maps in $\mathcal C$, and where a map $f\to g$ is a factorization $pfq=g$. Composition of $(p_1,q_1):f\to ...
4
votes
1answer
37 views

How to denote an 'atomic' morphism in category?

I want to distinguish between two disjoint classes of morphisms in a category: (1) those morphisms that are composed of other morphisms (other than identities) and could conceivably be factored into a ...
8
votes
2answers
129 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.
4
votes
1answer
169 views

What is a “foo” in category theory?

While browsing through several pages of nlab(mainly on n-Categories), I encountered the notion "foo" several times. However, there seems to be article on nlab about this notion. Is this some kind of ...
5
votes
1answer
178 views

What do I call a covariant functor which is a filtered colimit of representable functors?

Recall that a presheaf $C^{op} \to \text{Set}$ is pro-representable if it is a cofiltered limit of representable presheaves. The thing that represents it, roughly speaking, is a pro-object in $C$, ...
8
votes
1answer
337 views

Is “cofunctor” an accepted term for contravariant functors?

People are used to the prefix co- flipping arrows in a concept1, and I have seen people using cofunctor to mean a functor that flips arrows, i.e. that takes $A \to B$ to $FB \to FA$. I know this ...
3
votes
1answer
60 views

If monic, then *property*. Does the converse hold?

Theorem. Suppose $f : X \rightarrow Y$ is monic. Then for all $g : \bar{X} \rightarrow Y$ there exists at most one $h : \bar{X} \rightarrow X$ such that $f \circ h = g$. Question. Does the converse ...
2
votes
1answer
95 views

Direct product of groups in categorical terms

In the category of abelian groups $\mathsf{Ab}$ the coproduct is the "direct" product (only finite support), but the product is the "cartesian" product (may be infinite support). In the category of ...
7
votes
0answers
133 views

French translation of “well-powered” category

In order to write a report, I'm looking for a French translation of the term "well-powered category". Does anyone know the canonical term in French?
4
votes
1answer
121 views

Allegories in easy words?

1) What is, in easy words, the definiton of an allegory? 2) And when are allegories useful? What does it have to do with the category theory and categories? With the definiton of category, ...
3
votes
1answer
107 views

Mono's and Epi's in the category Rel?

Sorry to ask such a trivial question, but I can't find the answer anywhere. Question. What are the monomorphisms/epimorphisms in Rel? Furthermore, what's the standard terminology for describing ...
3
votes
1answer
63 views

What does it mean for a Category to have equalizers or/and pullbacks?

I know the definitions of what pullbacks and equalizars mean, but I don't know what it means that a given category $\mathfrak C$ has pullbacks or equalizers. Thanks
4
votes
0answers
137 views

Terminology Question: Precompose vs Compose?

I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
3
votes
1answer
84 views

Are distinctions in definitions of “finite” material in, eg, topology or measure theory?

There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set: (Richard ...