Given two categories $\mathcal{C},\mathcal{D}$, and a functor $F : \mathcal{C} \rightarrow \mathcal{D}$, we say that $F$
preserves a propery $P$ if whenever an object or arrow, $c$, of $\mathcal{C}$ has $P$, then so does $F(c)$.
reflects a property $P$ if whenever an object or arrow, $d = F(c)$, of $\mathcal{D}$ has $P$, then so does $c$.
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[math/Sets/] 1. Introduction to Sets;;Default.aspx?id=1;;;;[math/Sets/] 2. Relations on Sets;;Default.aspx?id=2;;;;[math/Misc/Magic/] 1. 1 = 0.999...;;Default.aspx?id=3;;;;[math/Misc/Oddities/] 1. Russell's Paradox;;Default.aspx?id=4;;;;[math/Misc/Oddities/] 2. The Orwell Group;;Default.aspx?id=20;;;;[math/Theories/Algebras/Group_Theory/] 1. Groups in General;;Default.aspx?id=5;;;;[math/Theories/Algebras/Monoid_Theory/] 1. Monoids in General;;Default.aspx?id=6;;;;[math/Theories/Algebras/Semigroup/] 1. Semigroups in General;;Default.aspx?id=7;;;;[math/Theories/Category_Theory/In_General/] 1. Definition;;Default.aspx?id=8;;;;[math/Theories/Category_Theory/In_General/] 2. Dual Category;;Default.aspx?id=16;;;;[math/Theories/Category_Theory/In_General/] 3. Mono-, Epi- and Isomorphisms;;Default.aspx?id=10;;;;[math/Theories/Category_Theory/In_General/] 4. Initial and Terminal Objects;;Default.aspx?id=14;;;;[math/Theories/Category_Theory/In_General/] 5. Products and Coproducts;;Default.aspx?id=15;;;;[math/Theories/Category_Theory/In_General/] 6. Exponentation and Cartesian Closed Categories;;Default.aspx?id=13;;;;[math/Theories/Category_Theory/Functors/] 1. Definition;;Default.aspx?id=9;;;;[math/Theories/Category_Theory/Functors/] 2. Full, Faithful and Embeddings;;Default.aspx?id=21;;;;[math/Theories/Category_Theory/Functors/] 3. Preserves and Reflects;;Default.aspx?id=22;;;;[math/Theories/Category_Theory/Functors/] 4. Contra-, Co- and Bivariant;;Default.aspx?id=23;;;;[math/Theories/Category_Theory/Functors/] 5. Co- and Contravariant Hom Functors;;Default.aspx?id=24;;;;[math/Theories/Category_Theory/Natural_Transformations/] 1. Definition and Examples;;Default.aspx?id=11;;;;[math/Theories/Category_Theory/Adjoints/] 1. Definition and Examples;;Default.aspx?id=25;;;;[math/Theories/Category_Theory/Applications/] 1. Transition Systems, Hoare Languages and Adjoints;;Default.aspx?id=26;;;;[math/Linear_Algebra/Misc/] 1. Cosine Similarity;;Default.aspx?id=12;;;;[math/Relations/] 1. Intro;;Default.aspx?id=17;;;;[math/Relations/] 2. Properties and Special Relations;;Default.aspx?id=18;;;;[math/Relations/] 3. Functions as relations;;Default.aspx?id=19;;;;[math/Functions/Elementary/] 1. Logarithmic;;Default.aspx?id=27;;;;[math/Functions/Elementary/] 2. Absoulte Value;;Default.aspx?id=28;;;;[math/Functions/Elementary/] 3. The Square Root;;Default.aspx?id=29;;;;