Kategoria przecinkowa – pojęcie używane w matematyce , w teorii kategorii .
Niech
A
,
{\displaystyle {\mathcal {A}},}
B
{\displaystyle {\mathcal {B}}}
C
{\displaystyle {\mathcal {C}}}
kategoriami , a
F
:
A
→
C
,
{\displaystyle F:{\mathcal {A}}\to {\mathcal {C}},}
G
:
B
→
C
{\displaystyle G:{\mathcal {B}}\to {\mathcal {C}}}
funktorami , tj.
A
→
F
C
←
G
B
.
{\displaystyle {\mathcal {A}}\xrightarrow {\;\;F\;\;} {\mathcal {C}}\xleftarrow {\;\;G\;\;} {\mathcal {B}}.}
Wówczas kategorią przecinkową nazywamy kategorię
(
F
↓
G
)
,
{\displaystyle (F\downarrow G),}
obiektami są trójki uporządkowane
(
A
,
f
,
B
)
,
{\displaystyle (A,f,B),}
A
∈
O
b
(
A
)
,
{\displaystyle A\in Ob({\mathcal {A}}),}
B
∈
O
b
(
B
)
{\displaystyle B\in Ob({\mathcal {B}})}
M
o
r
(
C
)
∋
f
:
F
(
A
)
→
G
(
B
)
;
{\displaystyle Mor({\mathcal {C}})\ni f:F(A)\to G(B);}
morfizmami
(
A
,
f
,
B
)
→
(
A
′
,
f
′
,
B
′
)
{\displaystyle (A,f,B)\to (A^{'},f^{'},B^{'})}
(
a
,
b
)
,
{\displaystyle (a,b),}
a
:
A
→
A
′
∈
M
o
r
(
A
)
,
{\displaystyle a:A\to A^{'}\in Mor({\mathcal {A}}),}
b
:
B
→
B
′
∈
M
o
r
(
B
)
,
{\displaystyle b:B\to B^{'}\in Mor({\mathcal {B}}),}
jest przemienny . Przy czym morfizmami tożsamościowymi są
id
(
A
,
f
,
B
)
=
(
id
A
,
id
B
)
,
{\displaystyle {\textrm {id}}_{(A,f,B)}=({\textrm {id}}_{A},{\textrm {id}}_{B}),}
(
a
′
,
b
′
)
∘
(
a
,
b
)
{\displaystyle (a^{'},b^{'})\circ (a,b)}
(
a
′
∘
a
,
b
′
∘
b
)
{\displaystyle (a^{'}\circ a,b^{'}\circ b)}
a
′
∘
a
,
b
′
∘
b
{\displaystyle a^{'}\circ a,b^{'}\circ b}
[1]
Jeżeli
C
=
A
{\displaystyle {\mathcal {C}}={\mathcal {A}}}
B
=
∗
,
{\displaystyle {\mathcal {B}}=*,}
B
{\displaystyle {\mathcal {B}}}
∗
{\displaystyle *}
G
(
∗
)
=
C
{\displaystyle G(*)=C}
C
∈
O
b
(
C
)
.
{\displaystyle C\in Ob({\mathcal {C}}).}
płatem kategorii
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle C}
C
/
C
.
{\displaystyle {\mathcal {C}}/C.}
Jeżeli zaś
C
=
B
{\displaystyle {\mathcal {C}}={\mathcal {B}}}
A
=
∗
,
{\displaystyle {\mathcal {A}}=*,}
kopłatem kategorii
C
{\displaystyle {\mathcal {C}}}
C
{\displaystyle C}
C
/
C
.
{\displaystyle C/{\mathcal {C}}.}
Gdy
A
=
B
=
C
{\displaystyle {\mathcal {A}}={\mathcal {B}}={\mathcal {C}}}
F
=
G
=
id
C
,
{\displaystyle F=G={\textrm {id}}_{\mathcal {C}},}
kategorią strzałkową i oznaczamy
C
→
{\displaystyle {\mathcal {C}}^{\to }}
[1]
↑ a b J. Adamek, H.Herrlich, G.E. Strecker: Abstract and Concrete Categories: The Joy of Cats . Dover Publications Inc., 2009, s. 43. 
