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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Example of a commutative square without a map between antidiagonal objects?

In an abelian category, can there be a commutative diagram of (vertical/horizontal) exact sequences $$ \require{AMScd} \begin{CD} 0 @>>> N @>>> M\\ @. @VVV @VVV \\ 0 @>>> X ...
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split epimorphism,equalizer,universal property

Let $g:B\to A$ be a split epimorphism with $f:A\to B$, $g\circ f=\operatorname{id}_A$. Why is $g$ a coequalizer of $f\circ g$ and $\operatorname{id}_B$? Commutativity is clear,but the universal ...
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75 views

conditions for an “exponential”

[More information in EDIT 2] If one defines an operation $\odot: V\times V\rightarrow V$ between the elements of a linear vector space, what properties should this operation have in order for a ...
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56 views

In additive category product and co-product over finite family of objects are isomorphism.

Let C be a additive category.Show that the co-product and product over finite family of objects are isomorphism.
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189 views

Two point topological space

Is there a standard name for the two point space with precisely one singleton being the only nontrivial open set? What are its most noteworthy categorical properties?
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48 views

Showing that a CCC with a zero object is the trivial category

Let $\mathcal{C}$ be a cartesian closed and assume that $0\cong 1$ (its initial object is the same as its terminal object). I want a detalied proof of the answer given here: ...
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80 views

Projective covers of graded modules.

I want to prove that there exist projective covers in the category of graded modules over an algebra. I am fairly new to "this" kind of mathematic and don't really know where to start: I found the ...
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62 views

Monoids, Semigroups, and a Reflective Subcategory.

The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity ...
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58 views

Natural numbers object via initial morphism

I assume that a natural number object (or see nLab) can be defined as an initial morphisms. (edit: as in the title, I ment initial morphism, not objects) $\hspace{1cm}$ Thoughts: Probably $X:=1$, ...
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58 views

Definition of cocone in category theory

I've been reading some basic category theory, and am slightly confused about the definition of a cocone. I've been looking at the notes here - ...
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57 views

Invertibility of Units/Counits

Suppose $\left \langle F,G,\eta ,\varepsilon \right \rangle$ is an adjunction. It is easy to show that if $G$ is full, then $G\varepsilon $ is invertible with inverse $\eta G$. But MacLane says that ...
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61 views

Not every over-under-category is cocomplete

Something is wrong between me and Hirschhorn: point (3) of this result seems to be false taken as it is: either I misunderstood something, or $(A\downarrow \mathcal{M}\downarrow B)$ is seldom ...
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52 views

Commutative diagrams with antiparallel arrows

A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean ...
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Do Natural transformations make 'God given' precise?

Often in mathematics, one encounters certain structures in which there are natural choices to be made. For example, there could be a 'god given' map relating two mathematical objects, in the sense ...
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58 views

Is a pushout of monics still monic?

I am sort of stuck with this problem and I have no idea if the answer is obvious and I am just too dumb to see it or if the question is really difficult. What i would like to prove is a particular ...
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32 views

What are the epis in Met?

I have an assignment to precisely describe epimorphisms and monomorphisms in Met (category whose objects are Metric spaces and whose morphisms are contractions). I have shown that Mono $\iff$ ...
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69 views

Free objects in the category of dg modules

Suppose that $A$ is a dg algebra, does the category of dg modules over $A$ where morphisms are degree zero maps that commute with differential have a free object ( in general)? I have been reading a ...
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A variant of projective objects?

Let $\mathcal{C}$ be an additive category. Is there a common name for objects $P \in \mathcal{C}$ with the property that $\hom(P,-) : \mathcal{C} \to \mathsf{Ab}$ is right exact, i.e. preserves all ...
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30 views

Universal Closure Operations

On page 227 in Borceux's Categorical Algebra I, he defines a universal closure operation on a finitely complete category $\mathscr B$ to be an operation on $\overline \square\colon \mathbf{Sub}(B)\to ...
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60 views

Zero object equivalent assertion

Let C be a category with zero object $0$. (i) Prove that for $A \in C$ the following assertions are equivalent: (a) A is a zero object; (b) $id_A$ is a zero morphism; (c) there is a monomorphism ...
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Does the category of directed systems have enough injectives?

Let $\mathscr{C}$ be an abelian category with enough injectives. Let $\mathscr{C}_A$ denote the abelian category of directed systems of objects of $\mathscr{C}$ indexed by $A$. Does $\mathscr{C}_A$ ...
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26 views

Categorically expressed property in $\textbf{Set}$ for $P(A)=\{17\}$

I'm reading Fokkinga's Gentle Introduction to Category Theory, of which page 12 asks to give categorically expressed properties (i.e., using the language of basic CT and existensial qualifiers) in ...
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63 views

2-category in HoTT: chapter 9 from the HoTT book

After reading Awodey's book, and the HoTT book (and numerous other entries on these topics), I am (ambitiously) trying to do the exercises after the category theory chapter. This concernes the ...
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35 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
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53 views

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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Continuous maps vs. open maps [duplicate]

When we study topology, we typically study topological spaces and continuous maps between them. From a categorical perspective, this is "wrong," because continuous maps are not the structure ...
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31 views

Topology making a family of functions optimal

I am trying to do a problem in Arbib's Category Theory book. Loosely rephrased: Let $\{(X_i,\tau_i)\}_I$ be a family of topological spaces, $X$ a set, and $\{f_i:X\to X_i\}_I$ a family of functions. ...
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Pushout with principal bundles

I am looking at the wikipedia page on reduction of the structure group for principal bundles (http://en.wikipedia.org/wiki/Reduction_of_the_structure_group) and at the beginning they introduce, for an ...
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160 views

Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V ...
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The closure of the graph of a certain composite in a topos.

Let $\mathcal{E}$ be an elementary topos with subobject classifier $\Omega$ and let $j\colon \Omega\to\Omega$ be a Lawvere-Tierney topology on it. Assume that, for an object $C$ of $\mathcal{E}$, each ...
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76 views

Question regarding adjoint functors

Suppose $B \rightarrow A$ is a morphism of rings. If $M$ is an $A$-module, one can create $M_B$ by considering it as a $B$-module. This gives a functor $\cdot_B: \mathrm{Mod}_A \rightarrow ...
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59 views

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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adjoint of forgetful functor related to localization

Let $A$ be a ring and $S$ a multiplicative subset of $A$ such that $1 \in S$. Let $G$ be the forgetful functor from $Mod_{S^{-1}A} \rightarrow Mod_A$. Taking an $S^{-1}A$-module N and consider it as ...
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69 views

How do I calculate adjoint functors to forgetful functors?

Suppose I have a forgetful functor $F:Ab\hookrightarrow Grp$ where we forget that we have commutativity. I'm trying to calculate the left and right adjoint functors, so for right adjoint, we have ...
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43 views

Confusion about pullback in $\sf{Set}$.

Given sets $A, B, C$ and set functions $f: A \to C$ and $g: B \to C$, everywhere I look seems to be telling me that the pullback is $A \times_C B = \left\{(a, b) \mid f(a) = f(b)\right\}$ (together ...
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The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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1answer
74 views

Some functorial maps $G\times G\rightarrow G$

Let $G$ be a group. Le diagonal map $\delta:G\rightarrow G\times G$ obviously gives a functorial morphism from the identity functor of $\mathbf{Grp}$ to the functor $P$ sending $G\mapsto G\times G$ ...
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35 views

Derived categories as homotopy categories of model categories

Given an abelian category A, is there a model structure on the category of complexes C(A) (or K(A) ("classical" homotopy category)) such that its homotopy category "is" the derived category D(A)?
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Where to study $2$-category theory?

Is there any place where I can read about $2$-categories? I am looking for a proper treatment - there is a section in Borceux's Handbook of Categorical Algebra, but it only sketches some parts of the ...
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17 views

Endofunctors on Cat preserving reflexive coequalizers

Is there any characterization of endofunctors on $\mathbf{Cat}$ preserving reflexive coequalizers?
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Must an epimorphism in abelian category have cokernel $0$?

Suppose $\mathcal{A}$ is an abelian category, that is an additive category with 1) a zero object, 2) all binary products and binary coproducts, 3) all kernels and cokernels. 4) monomorphisms are ...
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41 views

A seemingly simple fact about construction of maps proven categorically, i.e. by universal properties

If I have four sets $A,B,C,D$ and two maps $f_1 : A \to C$ and $f_2 : B \to D$, it is easy to find a unique map $f : A\times B \to C\times D$, namely $$ f(a,b) := (f_1(a), f_2(b)). $$ But now I want ...
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78 views

Transfinite composition and $\kappa$-categories

I wonder whether the following ideas make sense, and whether something in that direction has been written down somewhere; do you know a reference? The last definition is supposed to be a ...
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18 views

Morphisms of $\mathsf{Meas}$ and Dynamical Systems

The morphisms of a category $\mathsf{Meas}$ whose objects are measure spaces are defined to be equivalence classes of a.e-equal measurable maps that pull back null sets to null sets. Why is pulling ...
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81 views

Does the intersection of sets have a categorical interpretation?

My question is the title, really. I am wondering if the intersection of sets can be seen as a categorical construction on the objects of $\mathbf{Set}$.
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Monoidal categories and Generators

Let $\mathcal{C}$ be a Cocomplete Cowellpowered Monoidal category. Does $\mathcal{C}$ need to have a generator? I think it does not, but it seems hard to get a counter example.
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Adjoint functors for the power set monad

There is the power set functor, $T$, which gives raise to a monad: For a set $X$, we set $TX:=\mathcal P(X)$ and for $f:X\to Y$, we set $T(f):=S\mapsto f(S)$, where $f(S)$ denotes the direct image. ...
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Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
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question on fibred products

We are in a category where everything that follows exists. Is the fibred product $A \times_B (B \times_C D)$ isomorphic to $A \times_C D$ or to $A \times_C (A \times_B D)$?
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Question concerning the meaning of an equality sign in a commutative diagram

$\require{AMScd}$ I have the following question: Let $\mathscr{C}$ be a category, $X,Y,Z\in Ob(\mathscr{C}), \ f\in Mor(X,Y),\ g\in Mor(Y,Z)$ and $h\in Mor(X,Z)$. Question: What does the ...