Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

learn more… | top users | synonyms (2)

2
votes
0answers
55 views

When is Kleisli category regular?

Are there any known conditions on the monad $\mathbb{T}$, such that $\mathcal{C}_{\mathbb{T}}$ (in particular when $\mathcal{C} = \mathbf{Set}$) is regular?
5
votes
0answers
188 views

Generalization of analytic functors

A functor $F\colon \bf Sets\to Sets$ is said to be analytic if it results from the left Kan extension of a functor $f\colon \mathbf{Bij}(\mathbb N)\to \bf Sets$ (the "species" of the functors $F$) ...
2
votes
2answers
172 views

Difference between internal category and subcategory?

The category SET has an internal category, which is a small category with small objects and small morphisms, and that means that it's a subcollection of the collection of objects in SET. Is an ...
2
votes
0answers
127 views

What is the coimage of a ring homomorphism?

Is there a general way to find the coimage of a ring homomorphism? For example, for the canonical injection $\mathbb{Z}\rightarrow\mathbb{Q}$, the image is $\mathbb{Z}$ but the coimage is ...
2
votes
1answer
757 views

k-linear category

Let $C$ be a additive category and $k$ is a commutative ring. $C$ is called $k$-linear if the morphism sets $C(x,y)$ have the $k$-module structures for all $x,y\in Obj(C)$ and the compositions of ...
8
votes
7answers
582 views

Advice on self study of category theory

I'm very intrested in start to study category theory with aim to use this in algebraic geometry. I already took courses (besides the basics) on commutative algebra and general topology with a soft ...
12
votes
1answer
306 views

Are there useful categorical characterisations of the topological separation axioms?

Tietze's extension theorem states: If $X$ is a normal space, and $A$ a closed subspace. Then any continuous function to the reals $f:A\rightarrow R$ has an extension to $f':X\rightarrow R$ that is ...
8
votes
2answers
208 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.
3
votes
1answer
251 views

A example of a monoidal non symmetric category of $R$-bimodules

It is well know that if $R$ is a commutative ring, then the category of modules $_R\operatorname{Mod}$ is monoidal symmetric. (Edit. I wrote the category of $R$-bimodules, as F. Muro explained this ...
10
votes
2answers
325 views

Developing category theory inside ETCS

Trying to understand how Lawvere's [E]lementary [T]heory of the [C]ategory of [S]ets can be used as a foundation for mathematics alternative to ZFC, I am getting stuck with the question on how to ...
5
votes
0answers
47 views

Is there a diagrammatic language for bicartesian closed categories?

As in http://www.mscs.dal.ca/~selinger/papers/graphical.pdf but for a category more like a programming language with function, product, and sum types.
2
votes
1answer
83 views

When precisely can we replace quotient objects with subobjects in the definition of simple objects?

In a category with zero, a simple object is one that has only two quotients - itself and zero. Firstly - a point of confusion. The definition above says that quotient object requires a congruence, ...
3
votes
1answer
135 views

Does the nerve of a category preserve directed colimits?

The nerve $N(C)$ of a category $C$ is a simplicial set and defines a functor $$ N\colon\operatorname{Categories}\to \operatorname{sSets} $$ from the category of small categories to simplcial sets. It ...
7
votes
1answer
187 views

Is a filtered category necessarily (essentially) small?

There is a result, which I have heard is due to Grothendieck, which says that a left exact functor $F : C \to \text{Set}$ is a filtered colimit of representable functors provided that $C$ is ...
4
votes
1answer
72 views

Commuting squares in abelian categories

Here $A,B,C$ and $D$ are all objects in an Abelian category. $\require{AMScd} \begin{CD} A @> >> B @> >> C;\\ @VVV @VVV @VVV\\ D @> >>E @> >> F; \end{CD} $ The ...
4
votes
1answer
59 views

Existence of product in the category of pre-sheaves of abelian categories

Let $X$ be $Top(X)$ be the category of open sets of $X$ with inclusion maps as morphism. Let $\mathcal{C}$ be abelian category and $\mathcal{C}_x$ denote the category of contravariant functors from ...
1
vote
2answers
95 views

The full subcategory of representable controvariant functors

Let $\mathcal{C}$ be a category, let's denote by $\operatorname{Set}^{\mathcal{C}^{\operatorname{op}}}$ the category of presheaves of sets defined on $\mathcal{C}$ and natural transformations. I want ...
3
votes
1answer
152 views

Isomorphisms and natural transformations

Continuing my study through categories guided by Categories for the Working Mathematician, some questions about isomorphisms and natural transformations between bifunctors arose. The first exercise of ...
7
votes
1answer
167 views

A remark on triangulated categories and localizations in Kashiwara & Schapira's *Sheaves on Manifolds*

I'm having a little difficulty understanding the following remark in Kashiwara & Schapira's Sheaves on Manifolds: Since the term "null system" doesn't appear to be very common, here is the ...
2
votes
1answer
113 views

Checking if a function is injective

Let $\mathbb{C}$ be a small category, whose objects are thought of as "admissible worlds" and whose arrows as "temporal admissible developments". Let $X:\mathbb{C}^{\operatorname{op}}\rightarrow ...
4
votes
1answer
185 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 ...
6
votes
1answer
297 views

Why is the category of coherent sheaves not grothendieck?

Let $(X, \mathscr{O}_X)$ be a ringed space. It is well-known that the category $\mathbf{Mod}(\mathscr{O}_X)$ of $\mathscr{O}_X$-modules is a grothendieck abelian category (see e.g. Grothendieck's ...
5
votes
2answers
88 views

A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
5
votes
1answer
162 views

Homotopic maps in a directed system induce homotopic maps on colimit?

Let $(A_i,f_i)$ be a directed system of CW-complexes with colimit $A$. Further, let $g_i:A_i\to A_{i+1}$ be maps such that $g_{i+1}f_i=f_{i+1}g_i$ and $f_i\simeq g_i$. This might or might not be the ...
4
votes
1answer
238 views

Nicer proof that Yoneda embedding is continuous?

So if $\mathcal{C}$ is locally small, we have the Yoneda embedding $Y:\mathcal{C} \rightarrow [\mathcal{C}^{op},Sets ]$. This preserves all limits in $\mathcal{C}$, and a comment here: An application ...
6
votes
1answer
248 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 $C$, ...
5
votes
1answer
182 views

Why is there apparently no general notion of structure-homomorphism?

In model theory, one typically defines only embeddings of structures and isomorphisms, but I haven't seen a definition of general structure homomorphisms. Is there some particular reason behind that? ...
10
votes
1answer
138 views

Is there a universal property for the ultraproduct?

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i $ by the identification $x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in ...
1
vote
1answer
71 views

Derived functors - how is natural transformation between $L_0T$ and $T$ constructed?

For simplicity's sake, consider the categories $R\text{-Mod}, S\text{-Mod}$ of left $R$-modules and left $S$-modules, respectively, and let $\mathcal{F}$ be some precovering class in $R\text{-Mod}$. ...
8
votes
5answers
1k views

What is exactly the meaning of being isomorphic?

I know that the concept of being isomorphic depends on the category we are working in. So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or ...
8
votes
2answers
1k views

Definition of a monoid: clarification needed

I'm only in high school, so excuse my lack of familiarity with most of these terms! A monoid is defined as "an algebraic structure with a single associative binary operation and identity element." ...
6
votes
1answer
100 views

Recovering the structure of an object from its morphism:Yoneda Lemma

I've heard that Yoneda lemma informally states that one can recover the internal structure of an object by looking at the morphism coming out from that object. But this is not clear to me from the ...
4
votes
2answers
109 views

Problem understanding a proof in Model Categories by Hovey

I have serious problems understanding this proof from the book Model Categories, by Mark Hovey: Here's a list of things I don't understand: He's trying to prove the assertion by contradiction, ...
2
votes
2answers
161 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
132 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
3
votes
2answers
185 views

Computer algebra system for category theory

There are many different computer algebra systems which allows to perform computer aided computation in symbolic object like polynomials and function and also ordinals. Now being a category theory ...
1
vote
2answers
96 views

Monads on Set and their strength

I've been told that "every monad on Set has an unique strength". Is that true? (I can't seem to prove it) I did find a couple of hits googling, but they were mostly just statements without proofs. ...
3
votes
3answers
414 views

Definition of the Coproduct of Categories?

Let $\{\mathbf{C}_i~|~i \in I\}$ be a family of categories. What is $\coprod_{i\in I}\mathbf{C}_i$ ? Google and my regular sources have failed me. I might be able to anticipate (guess) the definition, ...
1
vote
1answer
59 views

Subobject classifiers as internalizations

I recently read the article on internalizations on nlab, but I am not quite sure what falls under that description. Is it fair to say, that subobjects are internalizations of subsets and that the ...
2
votes
1answer
55 views

Embedding into a morphism of distinguished triangles

Everything in this question happens in a triangulated category $\mathbf{D}$. I am trying to prove that in a diagram like this $$ ...
3
votes
1answer
76 views

Is this equality in a double category true?

Caveat: This is a utterly trivial question from a person who always learned to manipulate diagrams in a double category "from the ground"; I'll be glad even if you simply address me to any source ...
0
votes
2answers
210 views

is there a property of a category that is preserved by category isomorphism, but not equivalence?

I am trying to teach myself category theory and I am reading the section for Equivalence of categories. After some reading the question of "if there is a property of a category that is preserved by ...
10
votes
0answers
236 views

Transfinite horizontal composition

Suppose that you have two sequences $\{F_n\}$ and $\{G_n\}$ of endofunctors of $\bf A$ arising as strings of adjoints $\cdots\dashv F_{n-1}\dashv F_n\dashv F_{n+1}\dashv \cdots$, $\cdots\dashv ...
3
votes
2answers
129 views

Counter examples on Categories

I'm reading Categories for the Working Mathematician by Saunders Mac Lane. At the section 5 from chapter 1, for a fixed category, he claims that every arrow with right inverse, is epic (right ...
1
vote
1answer
169 views

Characterization of injective objects in abelian categories

In this link it is proved that in an abelian category $\mathcal C$ we have that $f:A\rightarrow B$ is mono iff the sequence $0\rightarrow A\rightarrow B$ is exact, where the arrow from $A$ to $B$ is ...
4
votes
1answer
77 views

Every morphism in Set is regular

I am trying to prove that every morphism in the category Set is regular, that is, that for every set-function $f:A\to B$ there exists a function $g:B\to A$ such that $f\circ g\circ f=f$. The ...
7
votes
2answers
206 views

Natural transformations in $\textbf{Set}$

I am trying to understand the concept of a natural transformation by considering the following example, an exercise from Mac Lane's Categories for the working mathematician (p. 18, ex. 1): Let $S$ ...
3
votes
2answers
76 views

isomorphic coequalizers

Suppose $e: B \rightarrow C$ is the coequalizer of two parallel morphisms $f,g; A \rightarrow B$. Show that if $e$ is monic then it is an isomorphism. I know that $e$ is epic. If $e$ is monic ...
-1
votes
1answer
39 views

Coequalizers of parallel morphisms

Show if $e : B \rightarrow C$ is the coequalizer of two parallel morphisms $f,g: A \rightarrow B$ and $w: W \rightarrow A$ is epic, then $e$ is the coequalizer of parallel morphisms $fw,gw: W ...
1
vote
2answers
97 views

morphisms on topological spaces

In the category of topological spaces: 1.) Show that a morphism is monic IFF it is injective 2.) Show that a morphism is epic IFF it is surjective 3.) Are there any morphisms that are monic and ...