Implication details
Claim: If a category is an elementary topos, then it has disjoint finite coproducts and has effective congruences and is epi-regular and is finitely cocomplete and is locally cartesian closed.
Proof: See Mac Lane–Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. IV.7.8; and Johnstone, Part A, Prop. 2.4.1 and Cor. 2.3.4.
Show 58 categories using this implication
- trivial category
- category of algebras
- category of finite sets and bijections
- delooping of the additive monoid of ordinal numbers
- category of sets equipped with a binary relation
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of commutative algebras
- category of commutative rings
- category of F(I)-sets
- category of filtered vector spaces
- category of finite sets
- category of finite sets of even cardinality
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of fields
- category of fields of characteristic zero
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of finitely generated groups
- category of Jónsson-Tarski algebras
- category of M-sets
- category of sets with a distinguished subset
- partially ordered set of natural numbers
- category of partially ordered sets without isolated points
- category of preordered sets
- category of quivers
- category of rings
- category of rngs
- category of sequences of sets
- category of connected sequences of sets
- category of sets
- category of set functions and commutative squares
- category of large families of sets
- category of sets with finite-to-one maps
- indiscrete category of sets
- category of pairs of sets
- category of empty-or-finite pairs of sets
- category of sheaves
- category of combinatorial species
- category of pointed topological spaces
- category of torsion-free abelian groups
- category of large families of vector spaces which are mostly zero
- category of finite Z-sets
- category of Z-sets
- real interval [0,1]
- category of simplicial sets
- walking commutative square
- walking composable pair
- walking fork
- walking isomorphism
- walking parallel pair
- walking span