Definition df-termc 49948

Description: Definition of the proper class (termcnex 50051) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49961). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 49990). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 49995). See also dftermc3 50006 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50006, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49966.

Followed directly from the definition, terminal categories are thin (termcthin 49952). The opposite category of a terminal category is "almost" itself (oppctermco 49980). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50015).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49976) and terminal objects (dftermo4 49977). The universal properties provide an alternate proof of initoeu1 17978, termoeu1 17985, initoeu2 17983, and termoeu2 49713. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 49991).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17654, 0thincg 49933, func0g 49564, 0funcg 49560, and initc 49566.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49947	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1541	. . . . . 6 class 𝑐
4		cbs 17179	. . . . . 6 class Base
5	3, 4	cfv 6498	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1541	. . . . . 6 class 𝑥
8	7	csn 4567	. . . . 5 class {𝑥}
9	5, 8	wceq 1542	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1781	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49892	. . 3 class ThinCat
12	10, 2, 11	crab 3389	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1542	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49949