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 ...

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1answer
194 views

What is meant by the Grothendieck group being the “best possible” construction of an abelian group from a commutative monoid?

On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid. What actually does this ...
4
votes
1answer
292 views

Small categories

Why $R$-Mod is a small category? There is a way to recognize small categories? For example Grp (i.e. category of all groups) is large because every set can be equiped with a group structure.
1
vote
1answer
302 views

Tensor product of sets

The cartesian product of two sets $A$ and $B$ can be seen as a tensor product. Are there examples for the tensor product of two sets $A$ and $B$ other than the usual cartesian product ? The context ...
2
votes
1answer
207 views

Retract of projective object is projective

An object $P$ in a category $\mathcal{C}$ is called projective if the functor $\mathcal{C}(P,-): \mathcal{C} \rightarrow Set$ preserves epimorphisms. Now I have to prove the following: Every retract ...
2
votes
1answer
126 views

Endomorphisms in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category generated by one element $X$ such that $End(X)=G$ where $G$ is a finite group. Is it true that, for any object $A \in \mathcal{C}$, $End(A)$ is ...
1
vote
0answers
86 views

Monoidal categories, but not in SET

We normally present the theory of categories in SET, that is, we define a category as a set of objects and a set of morphisms. If we do not present categories in SET, how do we present the abstract ...
4
votes
2answers
291 views

Did Structuralism influence the formulation of Category Theory?

Having only the a very cursory knowledge of Structuralism ( it's a movement generally held to have originated in linguistics, then moving on to philosophy & literature), there does appear to be ...
5
votes
1answer
52 views

Inherited Morita similar rings

Let $R$ and $S$ be Morita similar rings. If a ring $R$ with the following property: every right ideal is injective. How do I prove that the ring $S$ has this property? If a ring $R$ with the ...
-4
votes
1answer
243 views

Co/counter variancy of the Yoneda functor

What about proper co/counter variancy of the Yoneda embedding? $\operatorname{Hom}(C,-)$ or $\operatorname{Hom}(-,C)$? The Wikipedia seems to say something different than my books. Please explain ...
10
votes
1answer
652 views

Why is there no functor $\mathsf{Group}\to\mathsf{AbGroup}$ sending groups to their centers?

The category $\mathbf{Set}$ contains as its objects all small sets and arrows all functions between them. A set is "small" if it belongs to a larger set $U$, the universe. Let $\mathbf{Grp}$ be ...
3
votes
3answers
124 views

Intuition for Coconstant morphisms

A constant morphism $f \in \mathrm{Hom}(X,Y)$ is a morphism such that for any object $Z$ and any morphisms $g,h \in \mathrm{Hom}(Z,X)$, $f \circ g = f \circ h$. This is very easy to grasp and one can ...
1
vote
0answers
71 views

invert Grothendieck spectral sequence

I have 4 topoi $A,B,C,D$ (these are associated to abelian sheaves on some sites) and functors (which corresponds to some push-forward of sheaves) $F: A\rightarrow B$ $G: B \rightarrow C $ $H: A ...
2
votes
1answer
195 views

The axiom of choice and connected groupoids

Recall the two definitions of equivalence of categories: Definition 1. An equivalence of categories is a quadruplet $(F, G, \alpha, \beta)$, where $F : \mathbb{C} \to \mathbb{D}$ and $G : \mathbb{D} ...
4
votes
1answer
360 views

Equivalent definition of exactness of functor?

I'll use the following definition: (Def) A functor $F$ is exact if and only if it maps short exact sequences to short exact sequences. Now I'd like to prove the following (not entirely sure it's ...
-2
votes
2answers
228 views

Morphims with unique domain

A morphism $m$ of a category has the following property: No morphism (except of the identity morphism) of the category has codomain equal to the domain of $m$. In other words, $m$ cannot be composed ...
2
votes
0answers
89 views

Morphisms with an arbitrary number of objects

Is this structure familiar for you? It consists of a category $C$ a set $M$ a function ``$\operatorname{arity}$'' defined on $M$ a function $\operatorname{Obj}_m$ defined for every ...
4
votes
1answer
164 views

Do we implicitly consider model categories to be locally small?

Do we implicitly consider model categories to be locally small? I have the impression (but am not sure) that many references on model categories assume that all the categories are locally small, but ...
2
votes
2answers
427 views

Monomorphisms and fibrations are preserved by pullback

I just came across a strange property of morphisms that are preserved under pullbacks, and it made me wonder. Consider a model category $\mathcal{M}$. Because the fibrations are exactly the maps that ...
6
votes
2answers
125 views

What's the name of this operator?

Let $f,g$ be functions in $C^A$ and $C^B$ respectively. Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t. $f\boxtimes g(a,b)=(f(a),g(b))$ It seems not the tensor product, nor ...
7
votes
2answers
502 views

Understanding adjoint functors

To understand adjoint functors I tried to look at an example. Can you tell me if the following is correct? Before I give the example I'd like to recap the definition: Given two categories $C,D$ and ...
4
votes
0answers
120 views

When do we have the condition $\operatorname{ker} f = 0 \iff f$ monic?

Let $\mathcal{A}$ be a category. If $\mathcal{A}$ is pointed, i.e. has zero objects, then $f$ monic $\implies \operatorname{ker} f = 0$. If $\mathcal{A}$ is abelian, we have the equivalence $f$ monic ...
4
votes
1answer
293 views

Natural transformation

I'm trying to understand what a natural transformation is. To this end, I want to show the following: For each group $H$ the map $G \mapsto H \times G$ defines a functor $H \times -:\textbf{Grp} \to ...
0
votes
1answer
238 views

Sheaves in Grothendieck Topologies

Let $S$ be a scheme and the category of $S$-schemes be equipped with one of the standard Grothendieck topologies, say étale or fppf. Let $G \rightarrow H$ be a morphism of abelian sheaves on this ...
3
votes
1answer
149 views

Properties of functors

I want these functors to have the following properties, they seem a bit arbitrary though - so I was looking for sufficient "standard" properties of functors which imply them (such as full, faithful ...
4
votes
1answer
254 views

Is this the free abelian group functor?

Let $\mathbb{Z}(.) : \mathbf{Set} \to \mathbf{Ab}$ be the functor that assigns to any set $S$ the set of maps $\mathbb{Z}(S) := \{ z: S \to \mathbb{Z} \; | \; z(s)=0 \mbox{ for almost all } s \in S ...
3
votes
1answer
89 views

Fields as a reflective subcategory of integral domains?

A subcategory $\mathbf A$ is reflective subcategory of $\mathbf B$ if for every $B\in\mathbf B$ there exists an $A_B\in\mathbf A$ and a $\mathbf B$-morphism $r_B \colon A \to A_B$ such that: for any ...
7
votes
1answer
262 views

When is the pushout of a monic also monic?

Let $$\matrix{ A& \mathop{\longrightarrow}\limits^f &B\\ \Big\downarrow & & \Big\downarrow\\ C&\mathop{\longrightarrow}\limits_g &D }$$ Be a pushout diagram in a category ...
3
votes
1answer
80 views

When is a morphism of $S$-groupoids a monomorphism?

According to "Champs algébriques" by Laumon and Moret-Bailley, and $S$-groupoid is a category $\mathscr{X}$ and a functor $a: \mathscr{X} \to (\mathrm{Aff}/S)$, where $(\mathrm{Aff}/S)$ is the ...
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0answers
109 views

Adjoint prefunctors

I have a functor and a prefunctor (not a functor) "in the inverse direction". Can the notion of adjunction be generalized for prefunctors? I remind that a prefunctor is a functor without the ...
1
vote
1answer
69 views

All about (co)algebras for the identity functor

Any fact about these would be of interest. Has anyone seen an interesting structure constructed from a monad and comonad based on a single identity functor? I am working on a very small example just ...
2
votes
0answers
176 views

Applications of monads in general topology?

What are applications of monads in general topology? For example, for GT is important the notion of products, products are adjoints, so adjoints may be important for GT, but what's about monads?
0
votes
1answer
144 views

Compatibility of monoid and comonoid structures when monoidal product is a product

Let $\mathcal{C}$ be a category with finite products. Then $\mathcal{C}$ is a braided monoidal category with the product as the monoidal product and terminal object as the monoidal unit, and braiding ...
0
votes
2answers
111 views

Existence of non-identity natural transformations $\tau:F \to F $

Consider a functor $F:A\to B$. It is an object in category $B^A$. Question: is $B^A(F,F)$ a singleton? To put is in other words: do we have non-identity natural transformations $\tau:F \to F $?
2
votes
2answers
111 views

Does Bernstein theorem hold for models with elementary functions?

Bernstein theorem is a general pattern that occurs in many areas of mathematics (see the Wikipedia article for some examples). Does it hold for arbitrary models with elementary embeddings? To be more ...
6
votes
1answer
231 views

What are “Lazard” sheaves?

Early in Categories, Allegories, by Freyd and Scedrov (p.12, in the section on basic examples) there appears the following example: Let $\mathcal{LH}$ be the category whose objects are topological ...
4
votes
1answer
72 views

How do I show that the product of two monoidal objects is a monoidal object?

Let $(C,\otimes,1)$ be a braided monoidal category with braiding $\beta$. Given two monoidal objects, $A$ and $B$, with multiplication maps $\mu_A$ and $\mu_B$ and unit maps $e_A$ and $e_B$, I've ...
2
votes
2answers
980 views

How to prove the pullback lemma

I am new in category theory. I am trying to prove the well known fact that if you have a commutative diagram of the form □□, where each square is a pullback, then the whole diagram is a pullback too, ...
3
votes
1answer
349 views

why are subobjects defined to be equivalence classes of objects, instead of just objects?

In category theory, a subobject of object $A$ is defined to be an equivalence class of isomorphic monomorphisms into $A$. Does this seem weird to anyone else? Isn't it normal to allow something to be ...
2
votes
1answer
208 views

The Empty Category + The General Adjoint Functor Theorem

Let $\emptyset$ be the empty category. Consider the general adjoint functor theorem: it says that if $\mathcal{D}$ is locally small and complete then $G:\mathcal{D} \to \mathcal{C}$ has a left adjoint ...
1
vote
1answer
87 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 $F$? ...
4
votes
3answers
285 views

Uniqueness of Exponential Objects up to Isomorphism in any Category

I want to prove that for any pair of objects $a,b$ in a category $\mathcal{C}$, the exponential object $a^b$ of $a$ and $b$, if it exists, is unique up to isomorphism. It looks to be really simple, ...
5
votes
1answer
376 views

Excessive use of the Yoneda lemma

In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof: If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma. I understand ...
4
votes
2answers
415 views

Exponential objects in a cartesian closed category: $a^1 \cong a$

Hi I'm having problems with coming up with a proof for this simple property of cartesian closed categories (CCC) and exponential objects, namely that for any object $a$ in a CCC $C$ with an initial ...
3
votes
2answers
407 views

Understanding the Construction of the Quotient Category

I'm trying to understand, precisely, what a "quotient category" is. I've looked at several different definitions and they all seem to vary so I'm having a hard time nailing this concept down. A ...
2
votes
0answers
85 views

The Two Eilenberg-Moores

So, there is the Eilenberg-Moore spectral sequence, and there is (for any monad $(T,\mu,\eta)$ on a category $C$) the Eilenberg-Moore Category $C^T$ of $T$-algebras. The silly question, is the ...
1
vote
1answer
55 views

Is the morphism from product into fiber product a mono?

Let $C$ be a category with limits and $X\rightarrow Y$ a $C$-morphism. Is the induced morphism $X\times_YX\rightarrow X\times X$ a mono? In the category of sets, $X\times_YX$ is a subset of $X\times ...
14
votes
5answers
853 views

Concrete examples of 2-categories

I've been reading some of John Baez's work on 2-categories (eg here) and have been trying to visualize some of the constructions he gives. I'm interested in coming up with 'concrete' examples of ...
5
votes
2answers
240 views

Algebras over a field are flat - category theoretic proof?

Let $k$ be a field. Assume that you already know that the category $\mathrm{Alg}(k)$ of $k$-algebras (everything here is commutative and unital) has a coproduct $\sqcup$. But you don't know that this ...
6
votes
2answers
323 views

Is gcd the right adjoint of something?

In his answer link to the question whether $a|m$ and $a+1|m$ implies $a(a+1)|m$, Bill Dubuque takes a detour to derive the equality $$ \gcd(a,b)=ab/\mathrm{lcm}(a,b) $$ from the universal property of ...
2
votes
1answer
82 views

Direct products in subcategories

I have a several categories some of which are subcategories of others. I want to research properties of products in these categories but don't know where to start. How direct products in a category ...