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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When the empty family of arrows to an object is epimorphic, that object must be initial?

Is it true that when the empty family of arrows to an object $E$ in some category is epimorphic, that object $E$ must be the initial object $0$? This is a claim on page 433 (eq. 22) of Mac Lane and ...
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39 views

Projective Module equivalence [closed]

I'm trying to prove that given an $A-module$: $P$ The functor $Hom(P,-)$ is exact $\implies$ $\exists Q (A-module): P\oplus Q$ is free Can anyone guide me with some hints? It's the first time I'...
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Can we somehow use the functor $\mathbf{Set}(\mathbb{N},-)$ to define $\mathbb{N}$?

Hom functors can be used define coproducts in terms of products. In particular: $$\mathbf{Set}(A \sqcup B,X) \cong \mathbf{Set}(A,X) \times \mathbf{Set}(B,X)$$ To oversimplify a little: "a function ...
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Characterization of split mono- and split epimorphisms of algebras

Given a concrete category $(\mathcal{A}, U : \mathcal{A} \to \mathsf{Set})$ of universal algebras, obviously if a morphism $f$ in $\mathcal{A}$ is split mono / split epi, then so is $U(f)$. What is ...
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35 views

Pullback of a function along identity?

If $f:A\rightarrow B$ is a function, then $A\times _BB= \left\{ (a,b)\mid f(a)=b\right\}$. Isn't this the preimage of the image of $f$? I.e is $A\times_BB\cong f^\ast f_\ast(A)$? If working with ...
5
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81 views

In an algebraic category a morphism is a regular epi iff it is surjective

According to the nLab (see the 4th point under "Examples") in an "algebraic category" a morphism is a regular epi if and only, if it is surjective. Here a morphism $e$ is said to be surjective, if its ...
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Examples for the fact that a pullback of an epimorphism is not necessarily an epimorphism.

I'm reading in Borceux' book Basic Category Theory about pullbacks. It turns out that the pullback of an epimorphism is not necessarily an epimorphism. On the linked page, Borceux gives a ...
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33 views

Enough projectives and $F$ preserves limits implies $G$ preserves epi's.

Exercise: Let $\mathcal{C}, \mathcal{D}$ be categories, $G : \mathcal{C} \to \mathcal{D}$ and $F : \mathcal{D} \to \mathcal{C}$ an adjunction $F \dashv G$. Suppose $\mathcal{D}$ has enough projectives ...
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113 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
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1answer
54 views

post composition as a functor?

For every arrow $f$, we have a post composition functor $f_\ast$. However, one often uses the equality $(gf)_\ast=g_\ast\circ f_\ast$. I was wondering, what's the actual definition of this functor $(-)...
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Modal extensions (operators) for monoidal (categorical) logics

There is nice generalization of first order logic to monoidal (categorical) logics http://www.springer.com/us/book/9783642128202 which has recently been applied extensively as replacement for deontic ...
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Existence of pullback of fiber bundles from abstract nonsense?

Let $\mathsf C$ be a superextensive site with products. A trivial fiber bundle is a bundle $\pi:E\rightarrow B$ which is isomorphic to $\pi_1:B\times F\rightarrow B$ in $\mathsf{C}/B$. Let $\mathcal ...
2
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1answer
58 views

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
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How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left $B$-...
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41 views

Explicit unit/counit of inverse image/direct image adjunction.

Is there a nice explicit description for the unit and counit of the inverse image/direct image adjunction $f^{-1} \dashv f_*$ between sheaves of rings (and in the version $f^* \dashv f_*$ for $\...
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410 views

Why is the homotopy category actually important?

Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the ...
3
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91 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
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1answer
23 views

G adjoint iff initial object in $(D\downarrow G)$

$\def\D{\mathcal{D}}\def\C{\mathcal{C}}$ Let $G: \C \rightarrow \D$ be a functor. For each $D \in \D$ define the category $(D \downarrow G)$ which has as objects pairs $(C,g)$ with $C \in \C$ and $g: ...
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1answer
32 views

Functor is of the form Set(-,A)

Let $F: Set^{op} \rightarrow Set$ be a functor such that for corresponding functor $\overline{F}: Set \rightarrow Set^{op}$ we have $\overline{F} \dashv F$. With corresponding functor I mean that $F$ ...
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79 views

What is this structure involving a monad and a comonad?

Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can ...
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Functorial modifications of a topology

Let $S$ be a set. Then there are only two ways to attach functorialy a topology $\mathcal{T(S)}$ to it: The discrete and the trivial topology. Functorial means in this case that all maps $f \colon S \...
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1answer
85 views

The left adjoint to the forgetful functor $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ and Barr-Beck

Let $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ be the forgetful functor from $\mathbb{C}$-vector spaces to $\mathbb{R}$-vector spaces. I am trying to explicitly construct the left ...
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adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
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79 views

Weighted limits in the $Cat$-category of categories

What is a weighted limit in the $Cat$-category of categories, functors and natural transformations? I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
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69 views

The free cocompletion of a complete locally small category is complete

$\DeclareMathOperator{\colim}{colim}\newcommand{\cat}{\mathbf}\DeclareMathOperator{\Nat}{Nat}$I'm looking for a reference that talks about the free cocompletion $\hat{\cat C}$ of a (large) locally ...
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1answer
26 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...
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42 views

Is there a noteworthy example of an internal congruence in a category without binary products?

An internal relation in a category can be defined as a pair of jointly monic morphisms $r_1,r_2 : R \to A$ or equivalently as one monomorphism $r : R\to A\times A$, as long as the product $A\times A$ ...
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49 views

Concrete examples of epimorphism in the category of sets and partial functions

A epimorphism is a morphism $f : A \rightarrow B$ that is right-cancellative in the sense that, for all morphisms $g1, g2 : B \rightarrow X $ is true that \begin{equation} g_1\: o\: f = g_2\: o\: f ...
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1answer
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Monads and Monoids-as-categories

I'm trying to understand the definition of a monad as a monoid and to identify this structure in the implementation of monads in Hakell. One definition is that of a structure $(T,\eta,\mu)$ given by ...
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1answer
96 views

How to define a weighted cone?

Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X =...
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1answer
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Is there an extremal epi which is not epi, although binary products exist?

It is known, that in a category $\mathcal{A}$: strong epis are epi, if $\mathcal{A}$ has binary products extremal epis are epi, if $\mathcal{A}$ has equalizers all strong epis are extremal all ...
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Groupoid objects in the category of algebras

Can anyone give me some references where I could read about groupoid objects in the category of algebras? References about groupoid objects in other categories would also be welcome.
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“Spanning” a category

What does it mean for objects to "span" a category? I know what a span is but I'm not exactly sure what the phrase means. Does it just mean the category consists of the specified objects? An example ...
2
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1answer
38 views

Proving cardinality of coproduct presentation is unique without choice?

The definition of an extensive category immediately implies that given two coproduct decompositions indexed by sets of equal cardinality, if the coproduct objects are isomorphic compatibly with their ...
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1answer
80 views

What do you call such an object?

I would like to know if there is a name for an object $X$ in a (finitely complete and cocomplete) category $\mathcal{C}$ which has the following property: $X$ is non-empty and for every sub-object ...
3
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1answer
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In which situations is it possible to construct a coproduct this way?

Let $\left(F,U;\eta,\varepsilon\right):\mathcal{\mathcal{X}\rightharpoonup A}$ denote an adjunction where $U$ is faithful and where $\mathcal{X}$ is a category that has coproducts. You can think of $...
3
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1answer
48 views

Showing epis in $\mathbf{Grp}$ are surjective

I'm working through Maclane's Categories, and I got to exercise 5 of Section 5. In this question, the reader is asked to prove that all epis in $\mathbf{Grp}$ are surjective, and gives a sketch of ...
5
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1answer
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$F$ is an equivalence of categories implies that $F$ is fully faithful and essentially surjective

I read in wikipedia that: One can show that a functor $F : C → D$ yields an equivalence of categories if and only if $F$ is full, faithful and essnetially surjective. I'm trying to prove this but I ...
4
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1answer
55 views

Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
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1answer
89 views

Homology as categorification of Euler characteristic

I am trying to understand: "Thus, the homology of a manifold M can be seen, in a sense, as a categorification of its Euler characteristic: a more sophisticated and richly structured bearer of ...
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What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where ...
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1answer
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Finitely generated projective modules form exact category

I am looking for a proof of the fact that the category of f.g. projective modules over some commutative ring is exact. Actually, I would already be happy to know why this category is closed under ...
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The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category $\...
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Forgetful functor applied to a module

I try to find a left adjoint to the forgetful functor $U: R-Mod \longrightarrow Ab$. I considered a functor $F:Ab \longrightarrow R-Mod$ defined by $F(G)=Hom(U(R),G)$. I'm not so sure that in this ...
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1answer
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equivalence of definitions for connected objects?

For an extensive category, the following conditions are equivalent for an object $C$. The representable copresheaf of $C$ commutes with coproducts. The $C=X\amalg Y\implies X\text{ or }Y$ is $0$ and ...
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The benefit of writing Banach space theory in categorical language!

I was wondering if there exists a special benefit of writing Banach space theory in categorical language? I mean does there arise a hint of the existence of a connection with other mathematical field ...
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Slick Definition of the Category of Cartesian Closed Categories

I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ...
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Categorical Representation of the Set of All Strings

Let $A$ be a small preordered category. How would we define a preordered category $\mathcal A$ for all strings over $A$ (e.g., Kleene Closure) ordered by the subword order (Def'n $3.1$) $\leqslant$? ...
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cancellative property of coproducts in extensive categories?

In the final paragraph of the first section of this article, the following is written: Given an isomorphism $A+B\cong A+C$ compatible with the injections, one can construct an isomorphism $B\cong ...
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Exact adjoint functors of triangulated categories

Let $T$ and $S$ be triangulated categories. Let $F:T\rightarrow S$ and $G:S\rightarrow T$ be two adjoint functors. Assume that one of them is exact (i.e. sends exact triangles to exact triangles and ...