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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Are functors that are right-cancellable full, or do they have other characterizations? [duplicate]

In a former question it has become clear to me that a functor is left-cancellable if and only if it is injective on morphisms. This provides a nice characterization of monomorphisms in category ...
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Are functors that are left-cancellable necessarily injective on morphisms?

Let it be that $\mathcal C$ and $\mathcal D$ are categories and that $F:\mathcal C\rightarrow\mathcal D$ is a functor. If $F$ is injective on morphisms then it is easy to verify that it will be ...
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Set notation for generalized elements

I am currently reading Steve Awodey's book on category theory. On pg. 101 he uses set notation for generalized elements, namely $\{ a \mid f(a) = g(a) \}$ What does it mean in an arbitrary category?
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The image of an object under a subfunctor

Let $C$ be an additive category and $X$ is an object in $C$, $G$ is a functor in $(C^{op},Ab)$. $H$ is a subgroup of $G(X)$. Define $G_{H}(C)$ to be the set of all $a$ in $G(C)$ such that $G(f)(a)$ ...
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direct proof of the dual statement of the Yoneda lemma

The dual statement of the Yoneda lemma should read: Given any object $A$ in a locally small category $\mathsf{C}$ and any functor $F: \mathsf{C} \to \mathsf{Sets}$, we have an isomorphism ...
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If a finitely complete category has system of factorization (E,M) then the class M is stable under pullbacks.

We have to show that in this diagram ( see below) if $f \in$$\mathcal M$ then $g \in$$\mathcal M$. So $\forall e \in$ $\mathcal E$ we have to show $e \perp g$. I consider this diagram: But ...
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T-Algebras for a monad

Suppose $R$ is a ring with identity, $G:R$-$Mod\rightarrow Set$ is the forgetful functor and $F:Set\rightarrow R$-$Mod$ its left adjoint. I want to prove that the structure maps for the T-Algebras of ...
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Can we say anything about the relationship between these functors?

I am working with a category $\mathcal{C}$ and two functors $F:\mathcal{C}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ and $G:\mathcal{C}^{\operatorname{op}}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ where ...
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42 views

Does this property characterize monomorphisms?

Is requiring that $f:\mathrm{Hom}(A,B)$ is mono the same as requiring that the pullback $A\times_BA$ of $f$ along itself is isomorphic to $A$? In Sheaves in Geometry and Logic, I read the following ...
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Can two smooth categories be equivalent if their object manifolds aren't diffeomorphic or homotopy equivalent?

There is a category of "smooth categories", where the objects and the morphisms don't form sets, but manifolds (and there are some other conditions that I won't repeat here). Important examples are ...
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33 views

Modules in Morita Equivalence

In Method of Homological Algebra by Gelfand and Manin (Exercise 2.2.3). How are $\mathrm{Hom}_A(P,X)$ and $\mathrm{Hom}_B(P^*,Y)\,$ regarded as a $B$-module and $A$-module respectively?
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Regular representation, representability of the fiber functor, and hom-distributivity for Hilbert spaces

I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful ...
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58 views

Can a general version of the covariant powerset monad be derived from the universal property of power objects?

As the title asks, I'm wondering if one can generally squeeze a "covariant power object monad" out of a topos (following the usual example in $\mathcal{Set}$ with functor part the direct image ...
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51 views

Multiplication with category theory

Using category theory why is 3*2=6?. In book conceptual mathematics this is explained as : So there are 3 object 6,3 & 2 with two maps , level & shadow. There are 6 mapping from object 6 ...
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31 views

How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
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91 views

An equivalence of categories which looks like Voevodsky's Univalence Axiom

Let $\mathcal{C}$ be a category. Consider the full subcategory $\mathrm{Isom}(\mathcal{C})$ of $\mathrm{Mor}(\mathcal{C})$ whose objects are isomorphisms $A \xrightarrow{\cong} B$. It has a full ...
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39 views

Characterization of Projective Objects

In which categories is an object $P$ projective if and only if every short exact sequence ending with it splits? $$0\longrightarrow A\longrightarrow B\longrightarrow P \longrightarrow 0$$
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Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G ...
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50 views

why natural transformatoins are also called “morphisms” of functors?

I know a category can be described only with arrows, composition, domain and codomain operators.(without objects!) and a functor just a "morphism" between two categories, that is, it commutes with ...
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79 views

Why can I treat a category that isn't small as if it were?

I am working with a "combinatoric definition" of the zeroth and first homotopy groups for small categories. Let $C$ be a small category, the definitions are the following: The zeroth homotopy group ...
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49 views

Coproduct of bounded distributive lattices given as lattices of subsets

Let $X$ be a set. A lattice of subsets of $X$ is a subset of $\mathcal{P}(X)$ containing $\emptyset$ and $X$ and closed under finite intersection and finite union. Such a lattice is therefore a ...
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87 views

Perfect complexes and the derived category of a smooth projective variety

I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in $D^b(X)$ for $X$ smooth projective is then ...
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81 views

Reconciling three definitions of 'surjective'

A book on Category Theory for programmers defines surjection as such $f: A \to B$ that for any $y: T \to B$ there exists such $x: T \to A$ that $f\circ x = y.$ Then it illustrates it with ...
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Does this prove an equivalence of categories?

The definition I am working with (I know there is a stronger notion) states that a functor is an $\textbf{equivalence of categories}$ if it is fully faithful and essentially surjective. I was reading ...
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Intuition for dinatural and extranatural transformations

Conceptually, natural transformations make perfect sense. What is the intuition behind dinatural and extranatural transformations? Added: I'm looking for conceptual intuition, not something alone the ...
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38 views

How do I prove that $X$ is the pullback of this cospan?

Say we have a category $C$ with products and pullbacks. Let $X$ be an object of $C$. There's a canonical comonoid with multiplication $\Delta\colon X \to X\times X$, and it satisfies the ...
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Can we sensibly universally quantify over a category? (Not a question about size issues.)

You can get pretty far in mathematics without lambda abstraction. I guess this is because any formula of the form $\lambda^{x : X} f(x)=\lambda^{x : X}g(x)$ can always be replaced by the equivalent ...
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Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over ...
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30 views

Equalizers by pullbacks and products

I'm trying to solve exercise 5.6 in Steve Awodey's "Category Theory": Show that a category with pull-backs and products has equalizers as follows: given arrows $f, g: A \to B$, take the pullback ...
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44 views

Example of Hopf categories and conatural transformation

Let $\mathcal{C}$ be a category. It is well known how to internalize the notion of category. Let $(C_0,C_1)$ be an internal category, with source $s$, target $t$, composition $c$ and unit $e$. One can ...
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65 views

Categories of relations over a fixed category $\mathcal{C}$

Let $\bf{Set}$ be the category of sets and functions. We have an associated category $\bf{Set}_\bf{Rel}$, whose objects are also sets but whose morphisms are relations, i.e. a morphism ...
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Equivalence of categories involving graded modules and sheaves.

Let $S$ be a graded ring with $S_0=A$ a finitely generated $\mathbb{K}$-algebra and $S_1$ a finitely generated $A$-module. Let $M$ be a graded $S$-module and $\tilde{M}$ the corresponding sheaf on ...
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56 views

Cartesian Product characterization in Rel Category (Category of sets and relations) in simpler terms

The question here explains how a cartesian product of sets are specified. It is difficult for me to follow it; for example, I do not know about "symmetric monoidal category". Can anybody please ...
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Currying a continuous function

There is a mapping which I will call the "currying operator" $\hat\square:Z^{X\times Y}\to(Z^Y)^X$ which maps $f:X\times Y\to Z$ to $\hat f:X\to Z^Y$ defined as $x\mapsto y\mapsto f(x,y)$, or ...
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The first lemma in Auslander's functors and morphisms determined by objects

[lemma 1.1] Let $\mathcal{C}$ be a preadditive category. Suppose G is in ($\mathcal{C^{op}}$, $\mathcal{Ab}$). If for each $X$ in $\mathcal{C}$ we are given a subgroup $A_x$ of $G(X)$ such that ...
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1answer
49 views

An unexpected(?) way to identify a 2d vector space with its dual

Let $X$ be a finite-dimensional vector space over the field $\mathbb{R}$. Denote the dual of $X$ by $X^*$. Definition: Let us say that a (not necessarily linear) mapping $x \mapsto x^* : X \to ...
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Path to categorical realizability theory

I'm trying to understand the sorts of things found on this page: http://ncatlab.org/nlab/show/realizability In particular, I want to read Oosten's Realizability: An Introduction to the Categorical ...
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104 views

Is there a relationship between Turing's Halting theorem and Gödel Incompleteness

Turing's proof that a Halting oracle is impossible and Gödel's proof that and omega-consistent first order theory of arithmetic must be incomplete are similar in that they use self-referential ...
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Formalizing finitism in category theory

If we assume that finitism can be formalized by primitive recursive arithmetic (PRA), what category could it correspond to? In particular, which sort of a natural numbers object (NNO) may it contain? ...
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1answer
65 views

Is there a sleek categorical description of the obvious functor $(\mathrm{Mono}\,\mathbf{Set})^{\mathrm{op}} \rightarrow */\mathbf{Set}$?

Notation. Write $*/\mathbf{Set}$ for the category of pointed sets, and $F : \mathbf{Set} \rightarrow */\mathbf{Set}$ for the free functor (the "point-adjunction functor"). Given a category ...
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40 views

To what categorical concept is this proposition equivalent to…

In measure theory we have the following result (I know it's true for positive valued functions, so I'm taking a leap of faith assuming it's true for any measure space.) Proposition Let ...
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56 views

Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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1answer
56 views

If a functor is exact, does it always have an adjoint? If so, is the adjoint also exact?

For the first statement, if a functor is exact, can it admit both a left and right adjoint? (since it's both left and right exact). Under what conditions can these statements hold?
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Categories with basepoints from forgetful functor

The example of the forgetful functor: $$U: \text{Vect}_K \rightarrow Set$$ mapping the category of vector spaces over field $K$ to Set yields the category of elements consisting of based vector spaces ...
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Homotopy Extension Property as a pushout?

The usual diagram for the homotopy extension property is: where $i_t^X:X\rightarrow X\times I,x\mapsto(x,t)$. Isn't this the same as saying the following square is a pushout? $$\require{AMScd} ...
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Utility of the 2-Categorical Structure of $\mathsf{Top}$?

It's well known that $\mathsf{Top}$ is a 2-category with homotopy classes of homotopies as 2-arrows. I'm a bit afraid to ask this question, but what is the utility of this 2-categorical structure? ...
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72 views

How was the functorial factorization axiom “frequently misstated”?

In modern times, a model category is frequently required to have functorial factorizations of morphisms into (fibration/acyclic cofibration) and (acyclic fibration/cofibration). But the precise ...
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Is the pushforward measure a categorical-theoretic pushout?

Given two measurable spaces $(X,\mathscr{F}),(Y,\mathscr{G}),$ $f:X \to Y$ measurable and $\mu:\mathscr{F} \to [0,\infty)$ a measure, the pushforward of $f_*(\mu):\mathscr{G} \to [0, \infty)$ is ...
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Does “modular category” make sense without saying “abelian” or “linear”?

I know the term "modular category" only from representations of quantum groups, TQFTs and fusion (finitely semisimple linear) categories. There, a modular category is a ribbon fusion category where a ...
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Deformation retracts are closed under pushouts

I'm having trouble with the proof of Proposition 2.4.9 in Hovey's Model Categories. Proposition. Deformation retracts are closed under pushouts. Proof. Suppose we have a pushout diagram ...