I've been trying to figure out if the category of short exact sequences of abelian groups is abelian. It's clearly additive, and it has all of its kernels, but Im not sure about cokernels. I can't produce a solid counterexample, and don't know if it exists. Is the category abelian? And if not, what witnesses it's failure? I was thinking of the cokernel of a sequence like 0->Z->Z but I'm not sure (sorry for the poor formatting, I'm on my phone)
2026-03-26 22:16:11.1774563371
Is the category of short exact sequences of abelian groups abelian?
45 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in CATEGORY-THEORY
- (From Awodey)$\sf C \cong D$ be equivalent categories then $\sf C$ has binary products if and only if $\sf D$ does.
- Continuous functor for a Grothendieck topology
- Showing that initial object is also terminal in preadditive category
- Is $ X \to \mathrm{CH}^i (X) $ covariant or contravariant?
- What concept does a natural transformation between two functors between two monoids viewed as categories correspond to?
- Please explain Mac Lane notation on page 48
- How do you prove that category of representations of $G_m$ is equivalent to the category of finite dimensional graded vector spaces?
- Terminal object for Prin(X,G) (principal $G$-bundles)
- Show that a functor which preserves colimits has a right adjoint
- Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves
Related Questions in HOMOLOGICAL-ALGEBRA
- How does $\operatorname{Ind}^G_H$ behave with respect to $\bigoplus$?
- Describe explicitly a minimal free resolution
- $A$ - dga over field, then $H^i(A) = 0, i > 1$ implies $HH_i(A) = 0, i < -1$
- Tensor product $M\otimes_B Hom_B(M,B)$ equals $End_B(M)$, $M$ finitely generated over $B$ and projective
- Group cohomology of $\mathrm{GL}(V)$
- two maps are not homotopic equivalent
- Existence of adjugant with making given natural transformation be the counit
- Noetherian property is redundant?
- What is the monomorphism that forms the homology group?
- Rational points on conics over fields of dimension 1
Related Questions in ABELIAN-GROUPS
- How to construct a group whose "size" grows between polynomially and exponentially.
- $G$ abelian when $Z(G)$ is a proper subset of $G$?
- Invariant factor decomposition of quotient group of two subgroups of $\mathbb{Z}^n$.
- Computing Pontryagin Duals
- Determine the rank and the elementary divisors of each of the following groups.
- existence of subgroups of finite abelian groups
- Theorem of structure for abelian groups
- In the category of abelian groups the coequalizer $\text{Coker}(f, 0)$, $f: A \to B$ is simply $B/f(A)$.
- Commutator subgroup and simple groups
- Are there any interesting examples of functions on Abelian groups that are not homomorphisms?
Related Questions in EXACT-SEQUENCE
- Does every sequence of digits occur in one of the primes
- Linear transformation and Exact sequences
- Snake lemma and regular epi mono factorization
- Replacing terms of an exact sequence by quotients
- Module over integral domain, "Rank-nullity theorem", Exact Sequence
- Inclusion and quotient mappings in exact sequences
- Parsing the Bockstein morphism
- Short exact sequence on modules
- G-groups homomorphism regarding the subgroup fixed by G
- A problem about split exact sequences.
Related Questions in ABELIAN-CATEGORIES
- What is the monomorphism that forms the homology group?
- Injective objects in a category
- Category of complexes
- Snake lemma and regular epi mono factorization
- A question to Weibel’s IHA lemma 2.6.14 Part 2
- Why do the finitely generated subsheaves of a sheaf form a directed system?
- Supremum of a family of subobjects in an abelian category
- Opposite effective classes in a Grothendieck group
- Question about $\mbox{Ext}$ groups in abelian categories
- How to show that $\mathsf{Ab}$(Category of Abelian Groups) is an abelian category?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?