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 unit of a universal property is an isomorphism

Let $G \colon \mathbf B \to \mathbf A$ be a functor, and let $A \in \mathbf A$ be an object. A universal arrow from $A$ to $G$ can be described by an isomorphism of functors \begin{equation} \mathbf ...
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Relation between categorical operations (limits and co-limits)

Suppose I have a diagram $B \longleftarrow A \longrightarrow C$ in a category, and I execute a push-out operation and get $B \longrightarrow D \longleftarrow C$. If I execute a pull-back over $B ...
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What categorical property do these forgetful functors have in common?

Consider the following examples: The forgetful functor $U_1: \operatorname{Vect}_\mathbb{C} \to \operatorname{Vect}_\mathbb{R}$ The forgetful functor $U_2: \operatorname{Diff}^{\text{or}} \to ...
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Any deeper “duality” between non-zero-divisors and units of a ring?

I'm reading Aluffi's algebra book at the moment -- specifically, I'm on the introductory rings/modules chapter. I noticed two interesting pieces of information: in a (not necessarily commutative) ...
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38 views

Are there different combinatorial species with the same symmetry type?

First off: for my purposes, let $\sf B$ be the category of finite sets with bijections, and ${\sf B}_n$ the subcategory of sets with cardinality $n$, and define a combinatorial species to be a functor ...
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What is the name for a function whose codomain and domain are equal?

What do we call a function whose domain and co-domain are the same set? Edit: While i expressed my question in terms of functions, domains and codomains, i was actually interested in the most ...
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48 views

Explanation of proof in Representation Theory: A Homological Point of View

in the book Representation Theory: A Homological Point of View Proposition 3.1.18 Zimmerman proves that a cokernel is a colimit, but I can't understand his proof. He lets $\left((M_i)_{i \in ...
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Initial elements in Set and identity

By definition for every object there is at least one morphism - identity, so, there must be identity morphism for Set initial object - empty set. But no function can have empty set as codomain, so, ...
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Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

The Church/Turing Thesis that we all know and love asserts that every algorithmically computable function (in an informally characterized sense) is in fact recursive/Turing computable/lambda ...
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51 views

How to call a category with a single morphism between every two objects?

How to call a category where for every pair of objects $A, B$, there is a unique morphism $f\colon A\to B$? (A trivial category?)
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Interaction of functors and homology in abelian categories

I'm working on exercise 1.6.H.a) of Ravi Vakil's algebraic geometry course notes. I'm aware that a question was posted on the same topic before (Prove the FHHF theorem using as much abstract non-sense ...
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Specific case of tensor-hom adjunction

I'm currently working on a project, for which I need various bits of category theory which I've not seen much of before and do not know in detail, so I would like some confirmation (and probable ...
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Why is the “functor category” functor $(C,B)\mapsto B^{C}$ contravariant in $C$?

Good day everyone: I have been reading the book Categories for the Working Mathematicians and it is written that the functor category $B^{C}$ is itself a functor of the categories $B$ and $C$, ...
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Multicategories with out-arities

Basically, my question is: Why the emphasis on domains in the notion of multicategory? I will now give the formal framework to state it correctly. Passing from categories to multicategories ...
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57 views

Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$

Let $f:X\rightarrow Y$ be an arrow in $\mathsf{Set}$. $f$ induces three functors: $$\exists (f),\forall (f):\mathcal P(X)\rightarrow \mathcal P(Y),\; \mathsf I(f):\mathcal P(Y)\rightarrow \mathcal ...
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Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
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35 views

Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$

Let $P$ be a $\mathsf{Set}$-valued presheaf and let $f^\ast:\mathsf{PSh}(Y)\rightarrow \mathsf{PSh}(X)$ be the (topological) inverse image sheaf functor, defined on objects as the filtered colimit ...
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52 views

Does the concept of “cograph of a function” have natural generalisations / extensions?

First, definitions: The graph of a function $f : A \to B$ is a subset of $A \times B$, namely the set $\{(x,y) : x \in A, y \in B, f(x) = y\}$. The cograph of a function $f : A \to B$ is the ...
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Prove that the free group generated by two elements is a coproduct of the integers by itself in Grp

That's a problem from Algebra: Chapter $0$ by P. Aluffi, p.78, ex.5.6. One needs to prove that the group $F(\{x,y\})$ is a coproduct $\mathbb Z*\mathbb Z$ of $\mathbb Z$ by itself in the category ...
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30 views

Coproduct of rooted posets

The questions Currently, I'm working with "Category Theory" by Steve Awodey (which is quite readable, by the way). Their, on page 68 he gives an example for a coproduct, the coproduct in rooted ...
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Equal Categories [on hold]

Let $\infty$ be the "category" of all categories, where the objects are categories and the morphisms are functors. I am trying to motivate the definition of equal categories by doing it as follows. ...
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Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
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Understanding Tabulation in Rel Category

Tabulation in Allegories is a structure that is defined over the diagram $A \rightarrow B$. In a concrete Allegory, namely Rel category (Category of Sets and Relations) this seems to be the ...
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The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
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Monoids in Category Theory

I don't have a strong math background (engineering math) so I am at a bit of a disadvantage here but I have been trying to learn the broad strokes of Category Theory to help get a fuller picture of ...
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A functor preserves a product of $A$ and $B$ iff $F(A \times B) \cong F(A) \times F(B)$?

Let $F \colon \mathbf A \to \mathbf B$ be a functor, and let $A, B \in \mathbf A$. Assume that there exists a product $A \times B$, with projections $p \colon A \times B \to A$ and $q \colon A \times ...
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Category with zero

From "An Introduction to Ring Theory", Paul Cohn: "Let $\mathcal{A}$ be any category and define $\mathcal{A}''$ as the category obtained from $A$ by adjoining one object $Z$with a single morphism ...
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Why is the trivial group a zero object for the category of groups, but the empty set isn't a zero object for the category of sets? [duplicate]

I understand that the zero ring can't be a zero object for the category of rings, because in that case the 'arrows' are ring homomorphisms which, by definition, but maintain the unit. But in the ...
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42 views

Distributivity of pullbacks

If we consider morphisms $A\rightarrow C\leftarrow B$ in a category $\mathcal{C}$, then we denote their pullback by $A\leftarrow A\times_{C}B\rightarrow B$. The question is the following: is it true ...
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How is a connection the same as Morphism between different pullbacks to the first infinitesimal neighbourhood

This question arose from the first pages of Delignes Equations Differentielles. There he defines a connection on a vector bundle $V$ on an complex analytic space $X$ as the following Data: For all ...
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Properties preserved under equivalence of categories

I would like to ask about properties that are preserved under equivalence of categories. To be more specific, is it true that equivalences preserve limits? Why?
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Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
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2answers
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How can you take the dual of a category whose objects are Sets?

Let's say I have a category with two objects A {1, 2} B {3} I have the following morphisms ...
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Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
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Is there a special name for functors from a category C to a subcategory of C?

Is there a special name for functors from a category $C$ to a subcategory $S$ of $C$ where $S \ne C$? I'm using $\ne$ pretty informally above. I'd be happy with anything that reasonably captures the ...
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1answer
31 views

How should I think about morphism equality?

I am studying category theory, and I think I often struggle between the fuzzy lines between Set as a category and then the category theoretic abstractions. For example, take monomorphisms. I totally ...
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1answer
44 views

Is this correspondence covariant or controvariant?

I'm new to category theory and am trying a basic exercise. Is the correspondence from $S$ to $\mathcal{P}(S)$, which assigns to $f:S\rightarrow T$ the mapping $\mathcal{P}(S)\rightarrow ...
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Is every bijective pseudograph homomorphism a pseudograph isomorphism?

The term pseudograph describes a graph that may have parallel edges and loops. Formally this is a triple $G = (V,E,\delta)$ with $V,E$ sets and a map $\delta \colon E \to (V \times V)/\sim$, where ...
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29 views

Category of Sets and Bag-valued functions

I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. Now, It ...
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Contracted version of “isomorphic”

Had a look around and I can't find a word which acts as a contraction for "isomorphic" in the same way that "monic/epic" is a contraction of "monomorphic/epimorphic". For some reason this strikes me ...
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Group action on a category

Motivating example: We get a functor from the category of real vector spaces to the category of complex vector spaces by complexifying (i.e. tensoring over $\mathbb{R}$ with $\mathbb{C}$). Let ...
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Equivalent conditions for equivalence of categories (Proposition 7.26 in Awodey)

I'm trying to understand the proof of the following proposition in Steve Awodey's "Category Theory". Let $\mathbf{C}, \mathbf{D}$ be categories and let $F: \mathbf{C} \to \mathbf{D}$ be a functor. ...
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1answer
53 views

Can a contravariant functor be adjoint to a covariant one?

I am a bit confused about the definition of adjoint functors, since everywhere the definitions found (see example wikipedia https://en.wikipedia.org/wiki/Adjoint_functors) seem to not specify if we ...
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Direct (inductive) limit of groups

Let $(I,\prec)$ be a directed poset and $\{G_i\}_{i \in I}$ groups with group homomorphisms $f_{ij}:G_i \to G_j$ whenever $i \prec j$. Is is true that the direct limit of this system is given by $$ ...
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Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s

Motivated by this question, I want to find a complete set of relations for the ring of integer-valued polynomials, where the generators are the polynomials $\binom{x}{n}$ for $n\in \mathbb{N}$. The ...
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Compact objects in Ind-categories

Assume a category $\mathcal{D}$ admits small filtrant inductive limits. Then call an object $Y$ of $\mathcal{D}$ compact, if $\hom_{\mathcal{D}}(Y,\cdot )$ commutes with these small filtrant inductive ...
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Example of a forgetful functor that is not faithful.

In the forgetful functor Wikipedia article I read that "[Forgetful] Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may ...
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Category with coproducts generated by an endomorphism

Let's call a category with arbitrary coproducts a $\coprod$-category. A $\coprod$-functor is a functor which preserves coproducts. An example is $\mathsf{Set}$, and this is in fact the universal ...
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when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
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Monomorphisms and injectivity predicates

This is a curiosity question which struck me at a less-than-optimal moment; I apologize for not having thought much about it. Motivation. It is well-known that monomorphisms in a concrete category ...